If $f(x)≤x$ , then $f′(x)≤1$?












5












$begingroup$


I'm studying Calculus and having a trouble solving this question.



1) If $f(x)leq x$, then $f′(x)leq 1$ for all $x$?



2) What if $f(0)=0$, $f′(x)$ exists for all $x$?



I could easily find the counter example for 1) (Therefore it is false)



But I'm not sure about 2)



If $f(0)=0$ and $f′(x)$ exists for all $x$ &
$f(x)leq x$ , then $f′(x)leq 1$ for all $x$?



Please leave a comment if you don't mind :)






calculus derivatives







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  • $begingroup$
    The question is unclear: is the exercise ``consider a differentiable function $f$ such that $f(0)=0$ and $f(x)leq x$ for all $x$. Show that $f'(x)leq 1$ for all $x$.''?
    $endgroup$
    – b00n heT
    14 hours ago








  • 1




    $begingroup$
    I'm not sure that the new edits improved the question. I preferred the original version of the question, which gave a little more context.
    $endgroup$
    – littleO
    14 hours ago












  • $begingroup$
    Don't remove relevant information from your question!
    $endgroup$
    – user21820
    13 hours ago
















5












$begingroup$


I'm studying Calculus and having a trouble solving this question.



1) If $f(x)leq x$, then $f′(x)leq 1$ for all $x$?



2) What if $f(0)=0$, $f′(x)$ exists for all $x$?



I could easily find the counter example for 1) (Therefore it is false)



But I'm not sure about 2)



If $f(0)=0$ and $f′(x)$ exists for all $x$ &
$f(x)leq x$ , then $f′(x)leq 1$ for all $x$?



Please leave a comment if you don't mind :)






calculus derivatives







share|cite|improve this question









New contributor




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







$endgroup$












  • $begingroup$
    The question is unclear: is the exercise ``consider a differentiable function $f$ such that $f(0)=0$ and $f(x)leq x$ for all $x$. Show that $f'(x)leq 1$ for all $x$.''?
    $endgroup$
    – b00n heT
    14 hours ago








  • 1




    $begingroup$
    I'm not sure that the new edits improved the question. I preferred the original version of the question, which gave a little more context.
    $endgroup$
    – littleO
    14 hours ago












  • $begingroup$
    Don't remove relevant information from your question!
    $endgroup$
    – user21820
    13 hours ago














5












5








5


2



$begingroup$


I'm studying Calculus and having a trouble solving this question.



1) If $f(x)leq x$, then $f′(x)leq 1$ for all $x$?



2) What if $f(0)=0$, $f′(x)$ exists for all $x$?



I could easily find the counter example for 1) (Therefore it is false)



But I'm not sure about 2)



If $f(0)=0$ and $f′(x)$ exists for all $x$ &
$f(x)leq x$ , then $f′(x)leq 1$ for all $x$?



Please leave a comment if you don't mind :)






calculus derivatives







share|cite|improve this question









New contributor




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







$endgroup$




I'm studying Calculus and having a trouble solving this question.



1) If $f(x)leq x$, then $f′(x)leq 1$ for all $x$?



2) What if $f(0)=0$, $f′(x)$ exists for all $x$?



I could easily find the counter example for 1) (Therefore it is false)



But I'm not sure about 2)



If $f(0)=0$ and $f′(x)$ exists for all $x$ &
$f(x)leq x$ , then $f′(x)leq 1$ for all $x$?



Please leave a comment if you don't mind :)






calculus derivatives




calculus derivatives






share|cite|improve this question









New contributor




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











share|cite|improve this question









New contributor




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









share|cite|improve this question




share|cite|improve this question








edited 13 hours ago









user21820

39.4k543155




39.4k543155






New contributor




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asked 15 hours ago









Mighty QWERTYMighty QWERTY

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New contributor





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












  • $begingroup$
    The question is unclear: is the exercise ``consider a differentiable function $f$ such that $f(0)=0$ and $f(x)leq x$ for all $x$. Show that $f'(x)leq 1$ for all $x$.''?
    $endgroup$
    – b00n heT
    14 hours ago








  • 1




    $begingroup$
    I'm not sure that the new edits improved the question. I preferred the original version of the question, which gave a little more context.
    $endgroup$
    – littleO
    14 hours ago












  • $begingroup$
    Don't remove relevant information from your question!
    $endgroup$
    – user21820
    13 hours ago


















  • $begingroup$
    The question is unclear: is the exercise ``consider a differentiable function $f$ such that $f(0)=0$ and $f(x)leq x$ for all $x$. Show that $f'(x)leq 1$ for all $x$.''?
    $endgroup$
    – b00n heT
    14 hours ago








  • 1




    $begingroup$
    I'm not sure that the new edits improved the question. I preferred the original version of the question, which gave a little more context.
    $endgroup$
    – littleO
    14 hours ago












  • $begingroup$
    Don't remove relevant information from your question!
    $endgroup$
    – user21820
    13 hours ago
















$begingroup$
The question is unclear: is the exercise ``consider a differentiable function $f$ such that $f(0)=0$ and $f(x)leq x$ for all $x$. Show that $f'(x)leq 1$ for all $x$.''?
$endgroup$
– b00n heT
14 hours ago






$begingroup$
The question is unclear: is the exercise ``consider a differentiable function $f$ such that $f(0)=0$ and $f(x)leq x$ for all $x$. Show that $f'(x)leq 1$ for all $x$.''?
$endgroup$
– b00n heT
14 hours ago






1




1




$begingroup$
I'm not sure that the new edits improved the question. I preferred the original version of the question, which gave a little more context.
$endgroup$
– littleO
14 hours ago






$begingroup$
I'm not sure that the new edits improved the question. I preferred the original version of the question, which gave a little more context.
$endgroup$
– littleO
14 hours ago














$begingroup$
Don't remove relevant information from your question!
$endgroup$
– user21820
13 hours ago




$begingroup$
Don't remove relevant information from your question!
$endgroup$
– user21820
13 hours ago










3 Answers
3






active

oldest

votes


















3












$begingroup$

Draw the line $y=x$, and then draw any kind of squiggly function you want that stays below or touches the line. In particular, the function $f(x)=x-e^{-x}$ has $f'(x)gt1$ for all $x$, while $f(x)=x-{1over2}x^2$ satisfies $f(0)=0$ but $f'(x)gt1$ for $xlt0$.



Remark: The original version of the OP's question had two parts, with the condition $f(0)=0$ being added in the second part. The function $f(x)=x-e^{-x}$, of course, does not satisfy that condition.






share|cite|improve this answer











$endgroup$





















    4












    $begingroup$

    Hint: Consider $$f(x)=x-A sin^2 x $$
    for large $A$.






    share|cite|improve this answer









    $endgroup$





















      -2












      $begingroup$

      Let for example $f(x)=xsin x$, then $f(0)=0$ and for any real $x$ $f(x)leq x$. Then the derivative will look like this: $f'(x)=sin x+xcos x$ and so will exist for all $x$. Taking $x=2pi n$, where $n$ is positive integer we get $f'(x)=2pi n$, which evidently has no upper limitation.






      share|cite|improve this answer










      New contributor




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






      $endgroup$













      • $begingroup$
        $f(x)leq x$ fails for $x$ negative.
        $endgroup$
        – Wojowu
        14 hours ago











      Your Answer





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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      3












      $begingroup$

      Draw the line $y=x$, and then draw any kind of squiggly function you want that stays below or touches the line. In particular, the function $f(x)=x-e^{-x}$ has $f'(x)gt1$ for all $x$, while $f(x)=x-{1over2}x^2$ satisfies $f(0)=0$ but $f'(x)gt1$ for $xlt0$.



      Remark: The original version of the OP's question had two parts, with the condition $f(0)=0$ being added in the second part. The function $f(x)=x-e^{-x}$, of course, does not satisfy that condition.






      share|cite|improve this answer











      $endgroup$


















        3












        $begingroup$

        Draw the line $y=x$, and then draw any kind of squiggly function you want that stays below or touches the line. In particular, the function $f(x)=x-e^{-x}$ has $f'(x)gt1$ for all $x$, while $f(x)=x-{1over2}x^2$ satisfies $f(0)=0$ but $f'(x)gt1$ for $xlt0$.



        Remark: The original version of the OP's question had two parts, with the condition $f(0)=0$ being added in the second part. The function $f(x)=x-e^{-x}$, of course, does not satisfy that condition.






        share|cite|improve this answer











        $endgroup$
















          3












          3








          3





          $begingroup$

          Draw the line $y=x$, and then draw any kind of squiggly function you want that stays below or touches the line. In particular, the function $f(x)=x-e^{-x}$ has $f'(x)gt1$ for all $x$, while $f(x)=x-{1over2}x^2$ satisfies $f(0)=0$ but $f'(x)gt1$ for $xlt0$.



          Remark: The original version of the OP's question had two parts, with the condition $f(0)=0$ being added in the second part. The function $f(x)=x-e^{-x}$, of course, does not satisfy that condition.






          share|cite|improve this answer











          $endgroup$



          Draw the line $y=x$, and then draw any kind of squiggly function you want that stays below or touches the line. In particular, the function $f(x)=x-e^{-x}$ has $f'(x)gt1$ for all $x$, while $f(x)=x-{1over2}x^2$ satisfies $f(0)=0$ but $f'(x)gt1$ for $xlt0$.



          Remark: The original version of the OP's question had two parts, with the condition $f(0)=0$ being added in the second part. The function $f(x)=x-e^{-x}$, of course, does not satisfy that condition.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 14 hours ago

























          answered 14 hours ago









          Barry CipraBarry Cipra

          60.3k654126




          60.3k654126























              4












              $begingroup$

              Hint: Consider $$f(x)=x-A sin^2 x $$
              for large $A$.






              share|cite|improve this answer









              $endgroup$


















                4












                $begingroup$

                Hint: Consider $$f(x)=x-A sin^2 x $$
                for large $A$.






                share|cite|improve this answer









                $endgroup$
















                  4












                  4








                  4





                  $begingroup$

                  Hint: Consider $$f(x)=x-A sin^2 x $$
                  for large $A$.






                  share|cite|improve this answer









                  $endgroup$



                  Hint: Consider $$f(x)=x-A sin^2 x $$
                  for large $A$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 15 hours ago









                  user1337user1337

                  16.8k43592




                  16.8k43592























                      -2












                      $begingroup$

                      Let for example $f(x)=xsin x$, then $f(0)=0$ and for any real $x$ $f(x)leq x$. Then the derivative will look like this: $f'(x)=sin x+xcos x$ and so will exist for all $x$. Taking $x=2pi n$, where $n$ is positive integer we get $f'(x)=2pi n$, which evidently has no upper limitation.






                      share|cite|improve this answer










                      New contributor




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






                      $endgroup$













                      • $begingroup$
                        $f(x)leq x$ fails for $x$ negative.
                        $endgroup$
                        – Wojowu
                        14 hours ago
















                      -2












                      $begingroup$

                      Let for example $f(x)=xsin x$, then $f(0)=0$ and for any real $x$ $f(x)leq x$. Then the derivative will look like this: $f'(x)=sin x+xcos x$ and so will exist for all $x$. Taking $x=2pi n$, where $n$ is positive integer we get $f'(x)=2pi n$, which evidently has no upper limitation.






                      share|cite|improve this answer










                      New contributor




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






                      $endgroup$













                      • $begingroup$
                        $f(x)leq x$ fails for $x$ negative.
                        $endgroup$
                        – Wojowu
                        14 hours ago














                      -2












                      -2








                      -2





                      $begingroup$

                      Let for example $f(x)=xsin x$, then $f(0)=0$ and for any real $x$ $f(x)leq x$. Then the derivative will look like this: $f'(x)=sin x+xcos x$ and so will exist for all $x$. Taking $x=2pi n$, where $n$ is positive integer we get $f'(x)=2pi n$, which evidently has no upper limitation.






                      share|cite|improve this answer










                      New contributor




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






                      $endgroup$



                      Let for example $f(x)=xsin x$, then $f(0)=0$ and for any real $x$ $f(x)leq x$. Then the derivative will look like this: $f'(x)=sin x+xcos x$ and so will exist for all $x$. Taking $x=2pi n$, where $n$ is positive integer we get $f'(x)=2pi n$, which evidently has no upper limitation.







                      share|cite|improve this answer










                      New contributor




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









                      share|cite|improve this answer



                      share|cite|improve this answer








                      edited 14 hours ago









                      Max

                      881318




                      881318






                      New contributor




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









                      answered 15 hours ago









                      Alex KovalevskyAlex Kovalevsky

                      11




                      11




                      New contributor




                      Alex Kovalevsky is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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                      New contributor





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






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












                      • $begingroup$
                        $f(x)leq x$ fails for $x$ negative.
                        $endgroup$
                        – Wojowu
                        14 hours ago


















                      • $begingroup$
                        $f(x)leq x$ fails for $x$ negative.
                        $endgroup$
                        – Wojowu
                        14 hours ago
















                      $begingroup$
                      $f(x)leq x$ fails for $x$ negative.
                      $endgroup$
                      – Wojowu
                      14 hours ago




                      $begingroup$
                      $f(x)leq x$ fails for $x$ negative.
                      $endgroup$
                      – Wojowu
                      14 hours ago










                      Mighty QWERTY is a new contributor. Be nice, and check out our Code of Conduct.










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                      格利澤436b

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                      格利泽436b 太陽系外行星 太陽系外行星列表 艺术家笔下的格利泽436b 母恆星 母恆星 Gliese 436 星座 獅子座 赤經 ( α ) 11 h  42 m  11.0941 s [1] 赤緯 ( δ ) +26° 42′ 23.652″ [1] 距離 33.4 ly (10.2 pc) 光譜類型 M2.5 V [1] 軌道參數 半长轴 ( a ) 0.0291±0.0004 [2] AU 軌道離心率 ( e ) 0.150±0.012 [2] 公轉週期 ( P ) 2.643904±0.000005 [3] d (0.00723849 y) 軌道傾角 ( i ) 85.8 +0.21 −0.25 [3] ° 角距 ( θ ) 2.794 mas 近星點時間 ( T 0 ) 2,451,551.716 ±0.01 JD 半振幅 ( K ) 18.68±0.8 m/s 物理性质 质量 ( m ) 22.2±1.0 [2] M ⊕ 半徑 ( r ) 4.327±0.183 [2] [4] R ⊕ 密度 ( ρ ) 1.51 g cm -3 表面重力 ( g ) 1.18 g 溫度 ( T ) 712±36 [2] K 發現 發現時間 2004 發現者 巴特勒、沃格特、 馬西 et al. 發現方法 徑向速度、凌日法 發現地點 美國加利福尼亞州 發表論文 出版 其他名稱 Ross 905 b, GJ 436 b, [5] LTT 13213 b, GCTP 2704.10 b, LHS 310, AC+27:28217 b, Vyssotsky 616 b, HIP 57087 b, GEN# +9.80120068 b, LP 319-75 b, G 121-7 b, LSPM J1142+2642 b, 1RXS J114211.9+264328 b, ASCC 683818 b, G 147-68 b, UCAC2 41198281 b, BP...
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