How to prove that there is a differentiable function $f$ such that $[f(x)]^{153}+f(x)+x=0$ for all $x$












1












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Prove that there is a differentiable function $f$ such that $[f(x)]^{153}+f(x)+x=0$ for all $x$. Furthermore, find $f'$ in terms of $f$.



To me I just write $y$ instead of all $f(x)$ and find that $x=-(y^{153}+y)$ If I just derive that would that be the solution of this question? If not what should i do to answer it appropriately.










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  • $begingroup$
    Please write titles that describe the problem you wish to solve rather than the course title. This tutorial explains how to typeset mathematics on this site.
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    – N. F. Taussig
    Dec 8 '18 at 11:01










  • $begingroup$
    Can you advise me a description for this problem? Because I am a bit confused about the description.
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:04












  • $begingroup$
    I just placed the question in the title (while avoiding the command form, which is interpreted as rude on this site).
    $endgroup$
    – N. F. Taussig
    Dec 8 '18 at 11:07










  • $begingroup$
    Oh okay now it is obvious to me. Thank you for helping.
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:10
















1












$begingroup$


Prove that there is a differentiable function $f$ such that $[f(x)]^{153}+f(x)+x=0$ for all $x$. Furthermore, find $f'$ in terms of $f$.



To me I just write $y$ instead of all $f(x)$ and find that $x=-(y^{153}+y)$ If I just derive that would that be the solution of this question? If not what should i do to answer it appropriately.










share|cite|improve this question











$endgroup$












  • $begingroup$
    Please write titles that describe the problem you wish to solve rather than the course title. This tutorial explains how to typeset mathematics on this site.
    $endgroup$
    – N. F. Taussig
    Dec 8 '18 at 11:01










  • $begingroup$
    Can you advise me a description for this problem? Because I am a bit confused about the description.
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:04












  • $begingroup$
    I just placed the question in the title (while avoiding the command form, which is interpreted as rude on this site).
    $endgroup$
    – N. F. Taussig
    Dec 8 '18 at 11:07










  • $begingroup$
    Oh okay now it is obvious to me. Thank you for helping.
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:10














1












1








1





$begingroup$


Prove that there is a differentiable function $f$ such that $[f(x)]^{153}+f(x)+x=0$ for all $x$. Furthermore, find $f'$ in terms of $f$.



To me I just write $y$ instead of all $f(x)$ and find that $x=-(y^{153}+y)$ If I just derive that would that be the solution of this question? If not what should i do to answer it appropriately.










share|cite|improve this question











$endgroup$




Prove that there is a differentiable function $f$ such that $[f(x)]^{153}+f(x)+x=0$ for all $x$. Furthermore, find $f'$ in terms of $f$.



To me I just write $y$ instead of all $f(x)$ and find that $x=-(y^{153}+y)$ If I just derive that would that be the solution of this question? If not what should i do to answer it appropriately.







calculus algebra-precalculus






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share|cite|improve this question













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share|cite|improve this question








edited Dec 8 '18 at 11:06









N. F. Taussig

44.2k93356




44.2k93356










asked Dec 8 '18 at 10:57









HouseBTHouseBT

154




154












  • $begingroup$
    Please write titles that describe the problem you wish to solve rather than the course title. This tutorial explains how to typeset mathematics on this site.
    $endgroup$
    – N. F. Taussig
    Dec 8 '18 at 11:01










  • $begingroup$
    Can you advise me a description for this problem? Because I am a bit confused about the description.
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:04












  • $begingroup$
    I just placed the question in the title (while avoiding the command form, which is interpreted as rude on this site).
    $endgroup$
    – N. F. Taussig
    Dec 8 '18 at 11:07










  • $begingroup$
    Oh okay now it is obvious to me. Thank you for helping.
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:10


















  • $begingroup$
    Please write titles that describe the problem you wish to solve rather than the course title. This tutorial explains how to typeset mathematics on this site.
    $endgroup$
    – N. F. Taussig
    Dec 8 '18 at 11:01










  • $begingroup$
    Can you advise me a description for this problem? Because I am a bit confused about the description.
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:04












  • $begingroup$
    I just placed the question in the title (while avoiding the command form, which is interpreted as rude on this site).
    $endgroup$
    – N. F. Taussig
    Dec 8 '18 at 11:07










  • $begingroup$
    Oh okay now it is obvious to me. Thank you for helping.
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:10
















$begingroup$
Please write titles that describe the problem you wish to solve rather than the course title. This tutorial explains how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 8 '18 at 11:01




$begingroup$
Please write titles that describe the problem you wish to solve rather than the course title. This tutorial explains how to typeset mathematics on this site.
$endgroup$
– N. F. Taussig
Dec 8 '18 at 11:01












$begingroup$
Can you advise me a description for this problem? Because I am a bit confused about the description.
$endgroup$
– HouseBT
Dec 8 '18 at 11:04






$begingroup$
Can you advise me a description for this problem? Because I am a bit confused about the description.
$endgroup$
– HouseBT
Dec 8 '18 at 11:04














$begingroup$
I just placed the question in the title (while avoiding the command form, which is interpreted as rude on this site).
$endgroup$
– N. F. Taussig
Dec 8 '18 at 11:07




$begingroup$
I just placed the question in the title (while avoiding the command form, which is interpreted as rude on this site).
$endgroup$
– N. F. Taussig
Dec 8 '18 at 11:07












$begingroup$
Oh okay now it is obvious to me. Thank you for helping.
$endgroup$
– HouseBT
Dec 8 '18 at 11:10




$begingroup$
Oh okay now it is obvious to me. Thank you for helping.
$endgroup$
– HouseBT
Dec 8 '18 at 11:10










1 Answer
1






active

oldest

votes


















0












$begingroup$

HINT:



If $x = -[f(x)]^{153} - f(x)$,

then $g(x) = -x^{153} - x$ is the inverse function of $f(x)$.

Then if you prove $g(x)$ is differentiable, what does it tell about its inverse function ?






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Then it's inverse is differentiable?
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:10












  • $begingroup$
    Yes, you may look at exercise 12.14 for a sample problem here
    $endgroup$
    – rsadhvika
    Dec 8 '18 at 11:13












  • $begingroup$
    @HouseBT Unless $g'$ vanishes.
    $endgroup$
    – Michael Hoppe
    Dec 8 '18 at 11:14










  • $begingroup$
    Thank you so much ^^
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:15










  • $begingroup$
    Np :) hope you do see why $g$ is the inverse of $f$.. (Initially I had a problem visualizing..)
    $endgroup$
    – rsadhvika
    Dec 8 '18 at 11:17











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









0












$begingroup$

HINT:



If $x = -[f(x)]^{153} - f(x)$,

then $g(x) = -x^{153} - x$ is the inverse function of $f(x)$.

Then if you prove $g(x)$ is differentiable, what does it tell about its inverse function ?






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Then it's inverse is differentiable?
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:10












  • $begingroup$
    Yes, you may look at exercise 12.14 for a sample problem here
    $endgroup$
    – rsadhvika
    Dec 8 '18 at 11:13












  • $begingroup$
    @HouseBT Unless $g'$ vanishes.
    $endgroup$
    – Michael Hoppe
    Dec 8 '18 at 11:14










  • $begingroup$
    Thank you so much ^^
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:15










  • $begingroup$
    Np :) hope you do see why $g$ is the inverse of $f$.. (Initially I had a problem visualizing..)
    $endgroup$
    – rsadhvika
    Dec 8 '18 at 11:17
















0












$begingroup$

HINT:



If $x = -[f(x)]^{153} - f(x)$,

then $g(x) = -x^{153} - x$ is the inverse function of $f(x)$.

Then if you prove $g(x)$ is differentiable, what does it tell about its inverse function ?






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Then it's inverse is differentiable?
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:10












  • $begingroup$
    Yes, you may look at exercise 12.14 for a sample problem here
    $endgroup$
    – rsadhvika
    Dec 8 '18 at 11:13












  • $begingroup$
    @HouseBT Unless $g'$ vanishes.
    $endgroup$
    – Michael Hoppe
    Dec 8 '18 at 11:14










  • $begingroup$
    Thank you so much ^^
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:15










  • $begingroup$
    Np :) hope you do see why $g$ is the inverse of $f$.. (Initially I had a problem visualizing..)
    $endgroup$
    – rsadhvika
    Dec 8 '18 at 11:17














0












0








0





$begingroup$

HINT:



If $x = -[f(x)]^{153} - f(x)$,

then $g(x) = -x^{153} - x$ is the inverse function of $f(x)$.

Then if you prove $g(x)$ is differentiable, what does it tell about its inverse function ?






share|cite|improve this answer









$endgroup$



HINT:



If $x = -[f(x)]^{153} - f(x)$,

then $g(x) = -x^{153} - x$ is the inverse function of $f(x)$.

Then if you prove $g(x)$ is differentiable, what does it tell about its inverse function ?







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 8 '18 at 11:07









rsadhvikarsadhvika

1,7101228




1,7101228












  • $begingroup$
    Then it's inverse is differentiable?
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:10












  • $begingroup$
    Yes, you may look at exercise 12.14 for a sample problem here
    $endgroup$
    – rsadhvika
    Dec 8 '18 at 11:13












  • $begingroup$
    @HouseBT Unless $g'$ vanishes.
    $endgroup$
    – Michael Hoppe
    Dec 8 '18 at 11:14










  • $begingroup$
    Thank you so much ^^
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:15










  • $begingroup$
    Np :) hope you do see why $g$ is the inverse of $f$.. (Initially I had a problem visualizing..)
    $endgroup$
    – rsadhvika
    Dec 8 '18 at 11:17


















  • $begingroup$
    Then it's inverse is differentiable?
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:10












  • $begingroup$
    Yes, you may look at exercise 12.14 for a sample problem here
    $endgroup$
    – rsadhvika
    Dec 8 '18 at 11:13












  • $begingroup$
    @HouseBT Unless $g'$ vanishes.
    $endgroup$
    – Michael Hoppe
    Dec 8 '18 at 11:14










  • $begingroup$
    Thank you so much ^^
    $endgroup$
    – HouseBT
    Dec 8 '18 at 11:15










  • $begingroup$
    Np :) hope you do see why $g$ is the inverse of $f$.. (Initially I had a problem visualizing..)
    $endgroup$
    – rsadhvika
    Dec 8 '18 at 11:17
















$begingroup$
Then it's inverse is differentiable?
$endgroup$
– HouseBT
Dec 8 '18 at 11:10






$begingroup$
Then it's inverse is differentiable?
$endgroup$
– HouseBT
Dec 8 '18 at 11:10














$begingroup$
Yes, you may look at exercise 12.14 for a sample problem here
$endgroup$
– rsadhvika
Dec 8 '18 at 11:13






$begingroup$
Yes, you may look at exercise 12.14 for a sample problem here
$endgroup$
– rsadhvika
Dec 8 '18 at 11:13














$begingroup$
@HouseBT Unless $g'$ vanishes.
$endgroup$
– Michael Hoppe
Dec 8 '18 at 11:14




$begingroup$
@HouseBT Unless $g'$ vanishes.
$endgroup$
– Michael Hoppe
Dec 8 '18 at 11:14












$begingroup$
Thank you so much ^^
$endgroup$
– HouseBT
Dec 8 '18 at 11:15




$begingroup$
Thank you so much ^^
$endgroup$
– HouseBT
Dec 8 '18 at 11:15












$begingroup$
Np :) hope you do see why $g$ is the inverse of $f$.. (Initially I had a problem visualizing..)
$endgroup$
– rsadhvika
Dec 8 '18 at 11:17




$begingroup$
Np :) hope you do see why $g$ is the inverse of $f$.. (Initially I had a problem visualizing..)
$endgroup$
– rsadhvika
Dec 8 '18 at 11:17


















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