Determining properties of a polynomial $f$ satisfying $f(x^2)-xf(x) = x^4(x^2-1)$ for $x inBbb R^+$












2















Let $f$ be a polynomial satisfying $f(x^2)-xf(x) = x^4(x^2-1), x inBbb R^+$. Then which of the following is correct?



A) $f$ is an even function



B) $f$ is an odd function



C) $displaystylelim_{xto infty} frac{f(x)}{x^3}=1$



D) $displaystylelim_{xto infty} left(frac{f(x)}{x^2}-x right)$ exist and is equal to a non-zero quantity.




I have no idea what to do here.



Looking at the options, one thing I could guess is that the question wants us to find $f(x)$. After an analysis of few minutes, I could guess $f(x) = x^3$. But that gives me the answer as B), C). But the answer given is just C).



Any help would be appreciated.










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  • To ay that B) is true it is not enough to give one example. There may be other solutions which are not odd.
    – Kavi Rama Murthy
    Dec 20 at 6:23






  • 1




    Check the condition that says this polynomial is only satisfied when $x epsilon R^+$. So you don't know if the function is odd or not.
    – Sauhard Sharma
    Dec 20 at 6:27


















2















Let $f$ be a polynomial satisfying $f(x^2)-xf(x) = x^4(x^2-1), x inBbb R^+$. Then which of the following is correct?



A) $f$ is an even function



B) $f$ is an odd function



C) $displaystylelim_{xto infty} frac{f(x)}{x^3}=1$



D) $displaystylelim_{xto infty} left(frac{f(x)}{x^2}-x right)$ exist and is equal to a non-zero quantity.




I have no idea what to do here.



Looking at the options, one thing I could guess is that the question wants us to find $f(x)$. After an analysis of few minutes, I could guess $f(x) = x^3$. But that gives me the answer as B), C). But the answer given is just C).



Any help would be appreciated.










share|cite|improve this question









New contributor




Tony is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • To ay that B) is true it is not enough to give one example. There may be other solutions which are not odd.
    – Kavi Rama Murthy
    Dec 20 at 6:23






  • 1




    Check the condition that says this polynomial is only satisfied when $x epsilon R^+$. So you don't know if the function is odd or not.
    – Sauhard Sharma
    Dec 20 at 6:27
















2












2








2


1






Let $f$ be a polynomial satisfying $f(x^2)-xf(x) = x^4(x^2-1), x inBbb R^+$. Then which of the following is correct?



A) $f$ is an even function



B) $f$ is an odd function



C) $displaystylelim_{xto infty} frac{f(x)}{x^3}=1$



D) $displaystylelim_{xto infty} left(frac{f(x)}{x^2}-x right)$ exist and is equal to a non-zero quantity.




I have no idea what to do here.



Looking at the options, one thing I could guess is that the question wants us to find $f(x)$. After an analysis of few minutes, I could guess $f(x) = x^3$. But that gives me the answer as B), C). But the answer given is just C).



Any help would be appreciated.










share|cite|improve this question









New contributor




Tony is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












Let $f$ be a polynomial satisfying $f(x^2)-xf(x) = x^4(x^2-1), x inBbb R^+$. Then which of the following is correct?



A) $f$ is an even function



B) $f$ is an odd function



C) $displaystylelim_{xto infty} frac{f(x)}{x^3}=1$



D) $displaystylelim_{xto infty} left(frac{f(x)}{x^2}-x right)$ exist and is equal to a non-zero quantity.




I have no idea what to do here.



Looking at the options, one thing I could guess is that the question wants us to find $f(x)$. After an analysis of few minutes, I could guess $f(x) = x^3$. But that gives me the answer as B), C). But the answer given is just C).



Any help would be appreciated.







calculus limits functions even-and-odd-functions






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edited Dec 20 at 10:46









egreg

177k1484200




177k1484200






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asked Dec 20 at 6:18









Tony

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162




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Tony is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






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Check out our Code of Conduct.












  • To ay that B) is true it is not enough to give one example. There may be other solutions which are not odd.
    – Kavi Rama Murthy
    Dec 20 at 6:23






  • 1




    Check the condition that says this polynomial is only satisfied when $x epsilon R^+$. So you don't know if the function is odd or not.
    – Sauhard Sharma
    Dec 20 at 6:27




















  • To ay that B) is true it is not enough to give one example. There may be other solutions which are not odd.
    – Kavi Rama Murthy
    Dec 20 at 6:23






  • 1




    Check the condition that says this polynomial is only satisfied when $x epsilon R^+$. So you don't know if the function is odd or not.
    – Sauhard Sharma
    Dec 20 at 6:27


















To ay that B) is true it is not enough to give one example. There may be other solutions which are not odd.
– Kavi Rama Murthy
Dec 20 at 6:23




To ay that B) is true it is not enough to give one example. There may be other solutions which are not odd.
– Kavi Rama Murthy
Dec 20 at 6:23




1




1




Check the condition that says this polynomial is only satisfied when $x epsilon R^+$. So you don't know if the function is odd or not.
– Sauhard Sharma
Dec 20 at 6:27






Check the condition that says this polynomial is only satisfied when $x epsilon R^+$. So you don't know if the function is odd or not.
– Sauhard Sharma
Dec 20 at 6:27












3 Answers
3






active

oldest

votes


















7














Some questions have been raised in the comments about my answer below. A polynomial on $mathbb R$ which satisfies the given functional equation on $[0,infty)$ necessarily has the form $ax+x^{3}$ on that interval. But then it must have the same form on all of $mathbb R$. So I think the answer below is correct.



Look at the degree of the polynomial. If $f$ has degree $n$ then LHS has degree $2n$ and RHS has degree $6$. Hence $f$ is a polynomial of degree 3. The constant term is $0$ because $f(0)=0$. Show that the term in $x^{2}$ also must vanish because $xf(x)=f(x^{2})-x^{6}+x^{4}$ is a polynomial with only even powers of $x$. I guess you can take it from here. [All solutions are of the form $ax+x^{3}$]. Final anwser: A) and D) are false (the limit in D) exists but it is $0$); B) and C) are true.






share|cite|improve this answer























  • But the answer given is C).
    – Tony
    Dec 20 at 6:38










  • I double checked my solution and it looks right. I think the answer given is wrong.
    – Kavi Rama Murthy
    Dec 20 at 6:41










  • See Sauhard Sharma's comment. Can that be an explanation?
    – Tony
    Dec 20 at 6:50












  • @Kavi One doubt: If a function gas the set of domain restricted to positive reals only, then how can we judge abouts odd character. I mean we can't do $f(-x) =-f(x)$ (taking x as positive) as it is not permissible for $f(x)$ to obtain negative value of x as its domain. Am I right? Or, am I missing something obvious?
    – jayant98
    Dec 20 at 6:52






  • 1




    @jayant98 Please see the comments I have added to my answer. The equation is supposed to be satisfied only for $x$ positive but the polynomial is defined on the whole line and it is uniquely determined by its values on $[0,infty)$. So the question does make sense.
    – Kavi Rama Murthy
    Dec 20 at 7:25





















1














You don't need to find $f(x)$. For instance, statements A and B can be dealt with by considering
$$
f(x)=frac{f(x^2)-x^4(x^2-1)}{x} tag{*}
$$

and so
$$
f(-x)=frac{f(x^2)-x^4(x^2-1)}{-x}=-f(x)
$$

showing that $f$ is an odd function. One can object that (*) only holds for $x>0$, but the right-hand side is a polynomial, (implying $f(0)=0$) and if two polynomials agree on an infinite set they're equal.



If the degree of $f$ is $n$, then the degree of $f(x^2)$ is $2n$. By (*), the degree of $f(x^2)$ must be $6$, or the equality could not hold. Hence $n=3$.



The function $f(x)/x^3$ has finite limit $l$, owing to $deg f(x)=3$. Now
$$
frac{f(x^2)-xf(x)}{x^6}=frac{f(x^2)}{(x^2)^3}-frac{1}{x^2}frac{f(x)}{x^3}=frac{x^2-1}{x^2}=1-frac{1}{x^2}
$$

Hence $l=1$.



Similarly,
$$
frac{f(x)}{x^2}-x=frac{f(x^2)}{x^3}-x(x^2-1)-x=frac{f(x^2)}{x^3}-x^3=
xleft(frac{f(x^2)}{(x^2)^2}-x^2right)
$$

Now it's clear that statement D is false: the limit exists (finite or infinite), but if it's finite it must be $0$.



Therefore B and C are true. This allows us to find $f(x)$: it is a degree $3$ polynomial, with leading coefficient $1$ and no term of even degree. Hence we have
$$
f(x)=x^3+ax
$$

Apply the functional equation:
$$
f(x^2)-xf(x)=x^6+ax^2-x^4-ax^2=x^4(x^2-1)
$$

holds for every $x$. Therefore $a$ can be anything.






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    0














    To expand on @KaviRamaMurthy's answer:
    Let $f=sum^n a_kx^k$ where $a_nne 0$ for $xge 0$. Then
    $$sum a_kx^{2k}-sum a_kx^{k+1}=x^6-x^4$$
    $$sum a_kx^{2k}=sum a_kx^{k+1}+x^6-x^4$$
    So $2nin [0,n+1]cup{6}$ which means $nle 1$ or $n=3$. So
    $$f=b_0+b_1x+b_2x^2+b_3x^3$$ for $xge 0$. And then
    $$b_0+b_1x^2+b_2x^4+b_3x^6=b_0x+b_1x^2+b_2x^3+b_3x^4+x^6-x^4$$
    which means $b_3=1$ and then $b_2=0$: $b_1$ is unconstrained and $b_0=0$:
    $$f(x)=x^3+b_1x.$$



    The only way (B) can hold is if they meant




    for $xin R^+$, $f$ is equal to a polynomial such that... but $f$ need not be a polynomial on all of $R$.







    share|cite|improve this answer





















    • Any polynomial is defined on the whole real line (in fact the whole complex plane). The functional equation is given only on the positive real axis and that is enough to determine all solutions of the equation.
      – Kavi Rama Murthy
      Dec 20 at 7:28











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    3 Answers
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    active

    oldest

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    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    7














    Some questions have been raised in the comments about my answer below. A polynomial on $mathbb R$ which satisfies the given functional equation on $[0,infty)$ necessarily has the form $ax+x^{3}$ on that interval. But then it must have the same form on all of $mathbb R$. So I think the answer below is correct.



    Look at the degree of the polynomial. If $f$ has degree $n$ then LHS has degree $2n$ and RHS has degree $6$. Hence $f$ is a polynomial of degree 3. The constant term is $0$ because $f(0)=0$. Show that the term in $x^{2}$ also must vanish because $xf(x)=f(x^{2})-x^{6}+x^{4}$ is a polynomial with only even powers of $x$. I guess you can take it from here. [All solutions are of the form $ax+x^{3}$]. Final anwser: A) and D) are false (the limit in D) exists but it is $0$); B) and C) are true.






    share|cite|improve this answer























    • But the answer given is C).
      – Tony
      Dec 20 at 6:38










    • I double checked my solution and it looks right. I think the answer given is wrong.
      – Kavi Rama Murthy
      Dec 20 at 6:41










    • See Sauhard Sharma's comment. Can that be an explanation?
      – Tony
      Dec 20 at 6:50












    • @Kavi One doubt: If a function gas the set of domain restricted to positive reals only, then how can we judge abouts odd character. I mean we can't do $f(-x) =-f(x)$ (taking x as positive) as it is not permissible for $f(x)$ to obtain negative value of x as its domain. Am I right? Or, am I missing something obvious?
      – jayant98
      Dec 20 at 6:52






    • 1




      @jayant98 Please see the comments I have added to my answer. The equation is supposed to be satisfied only for $x$ positive but the polynomial is defined on the whole line and it is uniquely determined by its values on $[0,infty)$. So the question does make sense.
      – Kavi Rama Murthy
      Dec 20 at 7:25


















    7














    Some questions have been raised in the comments about my answer below. A polynomial on $mathbb R$ which satisfies the given functional equation on $[0,infty)$ necessarily has the form $ax+x^{3}$ on that interval. But then it must have the same form on all of $mathbb R$. So I think the answer below is correct.



    Look at the degree of the polynomial. If $f$ has degree $n$ then LHS has degree $2n$ and RHS has degree $6$. Hence $f$ is a polynomial of degree 3. The constant term is $0$ because $f(0)=0$. Show that the term in $x^{2}$ also must vanish because $xf(x)=f(x^{2})-x^{6}+x^{4}$ is a polynomial with only even powers of $x$. I guess you can take it from here. [All solutions are of the form $ax+x^{3}$]. Final anwser: A) and D) are false (the limit in D) exists but it is $0$); B) and C) are true.






    share|cite|improve this answer























    • But the answer given is C).
      – Tony
      Dec 20 at 6:38










    • I double checked my solution and it looks right. I think the answer given is wrong.
      – Kavi Rama Murthy
      Dec 20 at 6:41










    • See Sauhard Sharma's comment. Can that be an explanation?
      – Tony
      Dec 20 at 6:50












    • @Kavi One doubt: If a function gas the set of domain restricted to positive reals only, then how can we judge abouts odd character. I mean we can't do $f(-x) =-f(x)$ (taking x as positive) as it is not permissible for $f(x)$ to obtain negative value of x as its domain. Am I right? Or, am I missing something obvious?
      – jayant98
      Dec 20 at 6:52






    • 1




      @jayant98 Please see the comments I have added to my answer. The equation is supposed to be satisfied only for $x$ positive but the polynomial is defined on the whole line and it is uniquely determined by its values on $[0,infty)$. So the question does make sense.
      – Kavi Rama Murthy
      Dec 20 at 7:25
















    7












    7








    7






    Some questions have been raised in the comments about my answer below. A polynomial on $mathbb R$ which satisfies the given functional equation on $[0,infty)$ necessarily has the form $ax+x^{3}$ on that interval. But then it must have the same form on all of $mathbb R$. So I think the answer below is correct.



    Look at the degree of the polynomial. If $f$ has degree $n$ then LHS has degree $2n$ and RHS has degree $6$. Hence $f$ is a polynomial of degree 3. The constant term is $0$ because $f(0)=0$. Show that the term in $x^{2}$ also must vanish because $xf(x)=f(x^{2})-x^{6}+x^{4}$ is a polynomial with only even powers of $x$. I guess you can take it from here. [All solutions are of the form $ax+x^{3}$]. Final anwser: A) and D) are false (the limit in D) exists but it is $0$); B) and C) are true.






    share|cite|improve this answer














    Some questions have been raised in the comments about my answer below. A polynomial on $mathbb R$ which satisfies the given functional equation on $[0,infty)$ necessarily has the form $ax+x^{3}$ on that interval. But then it must have the same form on all of $mathbb R$. So I think the answer below is correct.



    Look at the degree of the polynomial. If $f$ has degree $n$ then LHS has degree $2n$ and RHS has degree $6$. Hence $f$ is a polynomial of degree 3. The constant term is $0$ because $f(0)=0$. Show that the term in $x^{2}$ also must vanish because $xf(x)=f(x^{2})-x^{6}+x^{4}$ is a polynomial with only even powers of $x$. I guess you can take it from here. [All solutions are of the form $ax+x^{3}$]. Final anwser: A) and D) are false (the limit in D) exists but it is $0$); B) and C) are true.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Dec 20 at 7:23

























    answered Dec 20 at 6:32









    Kavi Rama Murthy

    49.1k31854




    49.1k31854












    • But the answer given is C).
      – Tony
      Dec 20 at 6:38










    • I double checked my solution and it looks right. I think the answer given is wrong.
      – Kavi Rama Murthy
      Dec 20 at 6:41










    • See Sauhard Sharma's comment. Can that be an explanation?
      – Tony
      Dec 20 at 6:50












    • @Kavi One doubt: If a function gas the set of domain restricted to positive reals only, then how can we judge abouts odd character. I mean we can't do $f(-x) =-f(x)$ (taking x as positive) as it is not permissible for $f(x)$ to obtain negative value of x as its domain. Am I right? Or, am I missing something obvious?
      – jayant98
      Dec 20 at 6:52






    • 1




      @jayant98 Please see the comments I have added to my answer. The equation is supposed to be satisfied only for $x$ positive but the polynomial is defined on the whole line and it is uniquely determined by its values on $[0,infty)$. So the question does make sense.
      – Kavi Rama Murthy
      Dec 20 at 7:25




















    • But the answer given is C).
      – Tony
      Dec 20 at 6:38










    • I double checked my solution and it looks right. I think the answer given is wrong.
      – Kavi Rama Murthy
      Dec 20 at 6:41










    • See Sauhard Sharma's comment. Can that be an explanation?
      – Tony
      Dec 20 at 6:50












    • @Kavi One doubt: If a function gas the set of domain restricted to positive reals only, then how can we judge abouts odd character. I mean we can't do $f(-x) =-f(x)$ (taking x as positive) as it is not permissible for $f(x)$ to obtain negative value of x as its domain. Am I right? Or, am I missing something obvious?
      – jayant98
      Dec 20 at 6:52






    • 1




      @jayant98 Please see the comments I have added to my answer. The equation is supposed to be satisfied only for $x$ positive but the polynomial is defined on the whole line and it is uniquely determined by its values on $[0,infty)$. So the question does make sense.
      – Kavi Rama Murthy
      Dec 20 at 7:25


















    But the answer given is C).
    – Tony
    Dec 20 at 6:38




    But the answer given is C).
    – Tony
    Dec 20 at 6:38












    I double checked my solution and it looks right. I think the answer given is wrong.
    – Kavi Rama Murthy
    Dec 20 at 6:41




    I double checked my solution and it looks right. I think the answer given is wrong.
    – Kavi Rama Murthy
    Dec 20 at 6:41












    See Sauhard Sharma's comment. Can that be an explanation?
    – Tony
    Dec 20 at 6:50






    See Sauhard Sharma's comment. Can that be an explanation?
    – Tony
    Dec 20 at 6:50














    @Kavi One doubt: If a function gas the set of domain restricted to positive reals only, then how can we judge abouts odd character. I mean we can't do $f(-x) =-f(x)$ (taking x as positive) as it is not permissible for $f(x)$ to obtain negative value of x as its domain. Am I right? Or, am I missing something obvious?
    – jayant98
    Dec 20 at 6:52




    @Kavi One doubt: If a function gas the set of domain restricted to positive reals only, then how can we judge abouts odd character. I mean we can't do $f(-x) =-f(x)$ (taking x as positive) as it is not permissible for $f(x)$ to obtain negative value of x as its domain. Am I right? Or, am I missing something obvious?
    – jayant98
    Dec 20 at 6:52




    1




    1




    @jayant98 Please see the comments I have added to my answer. The equation is supposed to be satisfied only for $x$ positive but the polynomial is defined on the whole line and it is uniquely determined by its values on $[0,infty)$. So the question does make sense.
    – Kavi Rama Murthy
    Dec 20 at 7:25






    @jayant98 Please see the comments I have added to my answer. The equation is supposed to be satisfied only for $x$ positive but the polynomial is defined on the whole line and it is uniquely determined by its values on $[0,infty)$. So the question does make sense.
    – Kavi Rama Murthy
    Dec 20 at 7:25













    1














    You don't need to find $f(x)$. For instance, statements A and B can be dealt with by considering
    $$
    f(x)=frac{f(x^2)-x^4(x^2-1)}{x} tag{*}
    $$

    and so
    $$
    f(-x)=frac{f(x^2)-x^4(x^2-1)}{-x}=-f(x)
    $$

    showing that $f$ is an odd function. One can object that (*) only holds for $x>0$, but the right-hand side is a polynomial, (implying $f(0)=0$) and if two polynomials agree on an infinite set they're equal.



    If the degree of $f$ is $n$, then the degree of $f(x^2)$ is $2n$. By (*), the degree of $f(x^2)$ must be $6$, or the equality could not hold. Hence $n=3$.



    The function $f(x)/x^3$ has finite limit $l$, owing to $deg f(x)=3$. Now
    $$
    frac{f(x^2)-xf(x)}{x^6}=frac{f(x^2)}{(x^2)^3}-frac{1}{x^2}frac{f(x)}{x^3}=frac{x^2-1}{x^2}=1-frac{1}{x^2}
    $$

    Hence $l=1$.



    Similarly,
    $$
    frac{f(x)}{x^2}-x=frac{f(x^2)}{x^3}-x(x^2-1)-x=frac{f(x^2)}{x^3}-x^3=
    xleft(frac{f(x^2)}{(x^2)^2}-x^2right)
    $$

    Now it's clear that statement D is false: the limit exists (finite or infinite), but if it's finite it must be $0$.



    Therefore B and C are true. This allows us to find $f(x)$: it is a degree $3$ polynomial, with leading coefficient $1$ and no term of even degree. Hence we have
    $$
    f(x)=x^3+ax
    $$

    Apply the functional equation:
    $$
    f(x^2)-xf(x)=x^6+ax^2-x^4-ax^2=x^4(x^2-1)
    $$

    holds for every $x$. Therefore $a$ can be anything.






    share|cite|improve this answer




























      1














      You don't need to find $f(x)$. For instance, statements A and B can be dealt with by considering
      $$
      f(x)=frac{f(x^2)-x^4(x^2-1)}{x} tag{*}
      $$

      and so
      $$
      f(-x)=frac{f(x^2)-x^4(x^2-1)}{-x}=-f(x)
      $$

      showing that $f$ is an odd function. One can object that (*) only holds for $x>0$, but the right-hand side is a polynomial, (implying $f(0)=0$) and if two polynomials agree on an infinite set they're equal.



      If the degree of $f$ is $n$, then the degree of $f(x^2)$ is $2n$. By (*), the degree of $f(x^2)$ must be $6$, or the equality could not hold. Hence $n=3$.



      The function $f(x)/x^3$ has finite limit $l$, owing to $deg f(x)=3$. Now
      $$
      frac{f(x^2)-xf(x)}{x^6}=frac{f(x^2)}{(x^2)^3}-frac{1}{x^2}frac{f(x)}{x^3}=frac{x^2-1}{x^2}=1-frac{1}{x^2}
      $$

      Hence $l=1$.



      Similarly,
      $$
      frac{f(x)}{x^2}-x=frac{f(x^2)}{x^3}-x(x^2-1)-x=frac{f(x^2)}{x^3}-x^3=
      xleft(frac{f(x^2)}{(x^2)^2}-x^2right)
      $$

      Now it's clear that statement D is false: the limit exists (finite or infinite), but if it's finite it must be $0$.



      Therefore B and C are true. This allows us to find $f(x)$: it is a degree $3$ polynomial, with leading coefficient $1$ and no term of even degree. Hence we have
      $$
      f(x)=x^3+ax
      $$

      Apply the functional equation:
      $$
      f(x^2)-xf(x)=x^6+ax^2-x^4-ax^2=x^4(x^2-1)
      $$

      holds for every $x$. Therefore $a$ can be anything.






      share|cite|improve this answer


























        1












        1








        1






        You don't need to find $f(x)$. For instance, statements A and B can be dealt with by considering
        $$
        f(x)=frac{f(x^2)-x^4(x^2-1)}{x} tag{*}
        $$

        and so
        $$
        f(-x)=frac{f(x^2)-x^4(x^2-1)}{-x}=-f(x)
        $$

        showing that $f$ is an odd function. One can object that (*) only holds for $x>0$, but the right-hand side is a polynomial, (implying $f(0)=0$) and if two polynomials agree on an infinite set they're equal.



        If the degree of $f$ is $n$, then the degree of $f(x^2)$ is $2n$. By (*), the degree of $f(x^2)$ must be $6$, or the equality could not hold. Hence $n=3$.



        The function $f(x)/x^3$ has finite limit $l$, owing to $deg f(x)=3$. Now
        $$
        frac{f(x^2)-xf(x)}{x^6}=frac{f(x^2)}{(x^2)^3}-frac{1}{x^2}frac{f(x)}{x^3}=frac{x^2-1}{x^2}=1-frac{1}{x^2}
        $$

        Hence $l=1$.



        Similarly,
        $$
        frac{f(x)}{x^2}-x=frac{f(x^2)}{x^3}-x(x^2-1)-x=frac{f(x^2)}{x^3}-x^3=
        xleft(frac{f(x^2)}{(x^2)^2}-x^2right)
        $$

        Now it's clear that statement D is false: the limit exists (finite or infinite), but if it's finite it must be $0$.



        Therefore B and C are true. This allows us to find $f(x)$: it is a degree $3$ polynomial, with leading coefficient $1$ and no term of even degree. Hence we have
        $$
        f(x)=x^3+ax
        $$

        Apply the functional equation:
        $$
        f(x^2)-xf(x)=x^6+ax^2-x^4-ax^2=x^4(x^2-1)
        $$

        holds for every $x$. Therefore $a$ can be anything.






        share|cite|improve this answer














        You don't need to find $f(x)$. For instance, statements A and B can be dealt with by considering
        $$
        f(x)=frac{f(x^2)-x^4(x^2-1)}{x} tag{*}
        $$

        and so
        $$
        f(-x)=frac{f(x^2)-x^4(x^2-1)}{-x}=-f(x)
        $$

        showing that $f$ is an odd function. One can object that (*) only holds for $x>0$, but the right-hand side is a polynomial, (implying $f(0)=0$) and if two polynomials agree on an infinite set they're equal.



        If the degree of $f$ is $n$, then the degree of $f(x^2)$ is $2n$. By (*), the degree of $f(x^2)$ must be $6$, or the equality could not hold. Hence $n=3$.



        The function $f(x)/x^3$ has finite limit $l$, owing to $deg f(x)=3$. Now
        $$
        frac{f(x^2)-xf(x)}{x^6}=frac{f(x^2)}{(x^2)^3}-frac{1}{x^2}frac{f(x)}{x^3}=frac{x^2-1}{x^2}=1-frac{1}{x^2}
        $$

        Hence $l=1$.



        Similarly,
        $$
        frac{f(x)}{x^2}-x=frac{f(x^2)}{x^3}-x(x^2-1)-x=frac{f(x^2)}{x^3}-x^3=
        xleft(frac{f(x^2)}{(x^2)^2}-x^2right)
        $$

        Now it's clear that statement D is false: the limit exists (finite or infinite), but if it's finite it must be $0$.



        Therefore B and C are true. This allows us to find $f(x)$: it is a degree $3$ polynomial, with leading coefficient $1$ and no term of even degree. Hence we have
        $$
        f(x)=x^3+ax
        $$

        Apply the functional equation:
        $$
        f(x^2)-xf(x)=x^6+ax^2-x^4-ax^2=x^4(x^2-1)
        $$

        holds for every $x$. Therefore $a$ can be anything.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 20 at 11:26

























        answered Dec 20 at 9:55









        egreg

        177k1484200




        177k1484200























            0














            To expand on @KaviRamaMurthy's answer:
            Let $f=sum^n a_kx^k$ where $a_nne 0$ for $xge 0$. Then
            $$sum a_kx^{2k}-sum a_kx^{k+1}=x^6-x^4$$
            $$sum a_kx^{2k}=sum a_kx^{k+1}+x^6-x^4$$
            So $2nin [0,n+1]cup{6}$ which means $nle 1$ or $n=3$. So
            $$f=b_0+b_1x+b_2x^2+b_3x^3$$ for $xge 0$. And then
            $$b_0+b_1x^2+b_2x^4+b_3x^6=b_0x+b_1x^2+b_2x^3+b_3x^4+x^6-x^4$$
            which means $b_3=1$ and then $b_2=0$: $b_1$ is unconstrained and $b_0=0$:
            $$f(x)=x^3+b_1x.$$



            The only way (B) can hold is if they meant




            for $xin R^+$, $f$ is equal to a polynomial such that... but $f$ need not be a polynomial on all of $R$.







            share|cite|improve this answer





















            • Any polynomial is defined on the whole real line (in fact the whole complex plane). The functional equation is given only on the positive real axis and that is enough to determine all solutions of the equation.
              – Kavi Rama Murthy
              Dec 20 at 7:28
















            0














            To expand on @KaviRamaMurthy's answer:
            Let $f=sum^n a_kx^k$ where $a_nne 0$ for $xge 0$. Then
            $$sum a_kx^{2k}-sum a_kx^{k+1}=x^6-x^4$$
            $$sum a_kx^{2k}=sum a_kx^{k+1}+x^6-x^4$$
            So $2nin [0,n+1]cup{6}$ which means $nle 1$ or $n=3$. So
            $$f=b_0+b_1x+b_2x^2+b_3x^3$$ for $xge 0$. And then
            $$b_0+b_1x^2+b_2x^4+b_3x^6=b_0x+b_1x^2+b_2x^3+b_3x^4+x^6-x^4$$
            which means $b_3=1$ and then $b_2=0$: $b_1$ is unconstrained and $b_0=0$:
            $$f(x)=x^3+b_1x.$$



            The only way (B) can hold is if they meant




            for $xin R^+$, $f$ is equal to a polynomial such that... but $f$ need not be a polynomial on all of $R$.







            share|cite|improve this answer





















            • Any polynomial is defined on the whole real line (in fact the whole complex plane). The functional equation is given only on the positive real axis and that is enough to determine all solutions of the equation.
              – Kavi Rama Murthy
              Dec 20 at 7:28














            0












            0








            0






            To expand on @KaviRamaMurthy's answer:
            Let $f=sum^n a_kx^k$ where $a_nne 0$ for $xge 0$. Then
            $$sum a_kx^{2k}-sum a_kx^{k+1}=x^6-x^4$$
            $$sum a_kx^{2k}=sum a_kx^{k+1}+x^6-x^4$$
            So $2nin [0,n+1]cup{6}$ which means $nle 1$ or $n=3$. So
            $$f=b_0+b_1x+b_2x^2+b_3x^3$$ for $xge 0$. And then
            $$b_0+b_1x^2+b_2x^4+b_3x^6=b_0x+b_1x^2+b_2x^3+b_3x^4+x^6-x^4$$
            which means $b_3=1$ and then $b_2=0$: $b_1$ is unconstrained and $b_0=0$:
            $$f(x)=x^3+b_1x.$$



            The only way (B) can hold is if they meant




            for $xin R^+$, $f$ is equal to a polynomial such that... but $f$ need not be a polynomial on all of $R$.







            share|cite|improve this answer












            To expand on @KaviRamaMurthy's answer:
            Let $f=sum^n a_kx^k$ where $a_nne 0$ for $xge 0$. Then
            $$sum a_kx^{2k}-sum a_kx^{k+1}=x^6-x^4$$
            $$sum a_kx^{2k}=sum a_kx^{k+1}+x^6-x^4$$
            So $2nin [0,n+1]cup{6}$ which means $nle 1$ or $n=3$. So
            $$f=b_0+b_1x+b_2x^2+b_3x^3$$ for $xge 0$. And then
            $$b_0+b_1x^2+b_2x^4+b_3x^6=b_0x+b_1x^2+b_2x^3+b_3x^4+x^6-x^4$$
            which means $b_3=1$ and then $b_2=0$: $b_1$ is unconstrained and $b_0=0$:
            $$f(x)=x^3+b_1x.$$



            The only way (B) can hold is if they meant




            for $xin R^+$, $f$ is equal to a polynomial such that... but $f$ need not be a polynomial on all of $R$.








            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 20 at 6:54









            Bjørn Kjos-Hanssen

            2,064818




            2,064818












            • Any polynomial is defined on the whole real line (in fact the whole complex plane). The functional equation is given only on the positive real axis and that is enough to determine all solutions of the equation.
              – Kavi Rama Murthy
              Dec 20 at 7:28


















            • Any polynomial is defined on the whole real line (in fact the whole complex plane). The functional equation is given only on the positive real axis and that is enough to determine all solutions of the equation.
              – Kavi Rama Murthy
              Dec 20 at 7:28
















            Any polynomial is defined on the whole real line (in fact the whole complex plane). The functional equation is given only on the positive real axis and that is enough to determine all solutions of the equation.
            – Kavi Rama Murthy
            Dec 20 at 7:28




            Any polynomial is defined on the whole real line (in fact the whole complex plane). The functional equation is given only on the positive real axis and that is enough to determine all solutions of the equation.
            – Kavi Rama Murthy
            Dec 20 at 7:28










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