Prove that $x>y$

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Let $f$ be a strictly decreasing and strictly concave function. Assume that $x$ and $y$ satisfy the following conditions:



$f(x) + frac13xf'(x)=3$



$f(y) + yf'(y)=3$



Prove formally that $x>y$



I first solve for $x$ and $y$ and get



$x=frac3(3−f(x))f′(x)$



$y=frac3−f(y)f′(y)$



I'm not really sure how to proceed from here.



I asked this function about a month ago and received the response that such a function can't exist. If that is true (which I don't dispute) then it would be helpful to know how to loosen the initial constraints of the question such that it could be answered.




calculus proof-writing




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  • 1




    Can you add a link to your previous question with the answer that "... such a function can't exist" ?
    – Martin R
    Jul 19 at 18:06










  • @MartinR A strictly decreasing concave function on $(0,infty)$ cannot be a positive function (i.e., the range cannot be a subset of $(0,infty)$, as required in the problem statement).
    – Batominovski
    Jul 19 at 18:20











  • @Batominovski: I do not deny that. But if OP asks whether a previously received response is correct or not then he should at least add a link to that answer, so that we don't add another answer repeating the same arguments.
    – Martin R
    Jul 19 at 18:22











  • @MartinR It was deleted so I think only I can see it but here is the link in case you're able to see deleted Qs math.stackexchange.com/questions/2810441/prove-x-is-bigger-with
    – Frank Shmrank
    Jul 19 at 18:49










  • @Batominovski OK so if we relax the domain/range then can it be answerable?
    – Frank Shmrank
    Jul 19 at 18:50














up vote
-1
down vote

favorite












Let $f$ be a strictly decreasing and strictly concave function. Assume that $x$ and $y$ satisfy the following conditions:



$f(x) + frac13xf'(x)=3$



$f(y) + yf'(y)=3$



Prove formally that $x>y$



I first solve for $x$ and $y$ and get



$x=frac3(3−f(x))f′(x)$



$y=frac3−f(y)f′(y)$



I'm not really sure how to proceed from here.



I asked this function about a month ago and received the response that such a function can't exist. If that is true (which I don't dispute) then it would be helpful to know how to loosen the initial constraints of the question such that it could be answered.




calculus proof-writing




share|cite|improve this question

















  • 1




    Can you add a link to your previous question with the answer that "... such a function can't exist" ?
    – Martin R
    Jul 19 at 18:06










  • @MartinR A strictly decreasing concave function on $(0,infty)$ cannot be a positive function (i.e., the range cannot be a subset of $(0,infty)$, as required in the problem statement).
    – Batominovski
    Jul 19 at 18:20











  • @Batominovski: I do not deny that. But if OP asks whether a previously received response is correct or not then he should at least add a link to that answer, so that we don't add another answer repeating the same arguments.
    – Martin R
    Jul 19 at 18:22











  • @MartinR It was deleted so I think only I can see it but here is the link in case you're able to see deleted Qs math.stackexchange.com/questions/2810441/prove-x-is-bigger-with
    – Frank Shmrank
    Jul 19 at 18:49










  • @Batominovski OK so if we relax the domain/range then can it be answerable?
    – Frank Shmrank
    Jul 19 at 18:50












up vote
-1
down vote

favorite









up vote
-1
down vote

favorite











Let $f$ be a strictly decreasing and strictly concave function. Assume that $x$ and $y$ satisfy the following conditions:



$f(x) + frac13xf'(x)=3$



$f(y) + yf'(y)=3$



Prove formally that $x>y$



I first solve for $x$ and $y$ and get



$x=frac3(3−f(x))f′(x)$



$y=frac3−f(y)f′(y)$



I'm not really sure how to proceed from here.



I asked this function about a month ago and received the response that such a function can't exist. If that is true (which I don't dispute) then it would be helpful to know how to loosen the initial constraints of the question such that it could be answered.




calculus proof-writing




share|cite|improve this question













Let $f$ be a strictly decreasing and strictly concave function. Assume that $x$ and $y$ satisfy the following conditions:



$f(x) + frac13xf'(x)=3$



$f(y) + yf'(y)=3$



Prove formally that $x>y$



I first solve for $x$ and $y$ and get



$x=frac3(3−f(x))f′(x)$



$y=frac3−f(y)f′(y)$



I'm not really sure how to proceed from here.



I asked this function about a month ago and received the response that such a function can't exist. If that is true (which I don't dispute) then it would be helpful to know how to loosen the initial constraints of the question such that it could be answered.





calculus proof-writing





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 22 at 20:26
























asked Jul 19 at 18:02









Frank Shmrank

279112




279112







  • 1




    Can you add a link to your previous question with the answer that "... such a function can't exist" ?
    – Martin R
    Jul 19 at 18:06










  • @MartinR A strictly decreasing concave function on $(0,infty)$ cannot be a positive function (i.e., the range cannot be a subset of $(0,infty)$, as required in the problem statement).
    – Batominovski
    Jul 19 at 18:20











  • @Batominovski: I do not deny that. But if OP asks whether a previously received response is correct or not then he should at least add a link to that answer, so that we don't add another answer repeating the same arguments.
    – Martin R
    Jul 19 at 18:22











  • @MartinR It was deleted so I think only I can see it but here is the link in case you're able to see deleted Qs math.stackexchange.com/questions/2810441/prove-x-is-bigger-with
    – Frank Shmrank
    Jul 19 at 18:49










  • @Batominovski OK so if we relax the domain/range then can it be answerable?
    – Frank Shmrank
    Jul 19 at 18:50












  • 1




    Can you add a link to your previous question with the answer that "... such a function can't exist" ?
    – Martin R
    Jul 19 at 18:06










  • @MartinR A strictly decreasing concave function on $(0,infty)$ cannot be a positive function (i.e., the range cannot be a subset of $(0,infty)$, as required in the problem statement).
    – Batominovski
    Jul 19 at 18:20











  • @Batominovski: I do not deny that. But if OP asks whether a previously received response is correct or not then he should at least add a link to that answer, so that we don't add another answer repeating the same arguments.
    – Martin R
    Jul 19 at 18:22











  • @MartinR It was deleted so I think only I can see it but here is the link in case you're able to see deleted Qs math.stackexchange.com/questions/2810441/prove-x-is-bigger-with
    – Frank Shmrank
    Jul 19 at 18:49










  • @Batominovski OK so if we relax the domain/range then can it be answerable?
    – Frank Shmrank
    Jul 19 at 18:50







1




1




Can you add a link to your previous question with the answer that "... such a function can't exist" ?
– Martin R
Jul 19 at 18:06




Can you add a link to your previous question with the answer that "... such a function can't exist" ?
– Martin R
Jul 19 at 18:06












@MartinR A strictly decreasing concave function on $(0,infty)$ cannot be a positive function (i.e., the range cannot be a subset of $(0,infty)$, as required in the problem statement).
– Batominovski
Jul 19 at 18:20





@MartinR A strictly decreasing concave function on $(0,infty)$ cannot be a positive function (i.e., the range cannot be a subset of $(0,infty)$, as required in the problem statement).
– Batominovski
Jul 19 at 18:20













@Batominovski: I do not deny that. But if OP asks whether a previously received response is correct or not then he should at least add a link to that answer, so that we don't add another answer repeating the same arguments.
– Martin R
Jul 19 at 18:22





@Batominovski: I do not deny that. But if OP asks whether a previously received response is correct or not then he should at least add a link to that answer, so that we don't add another answer repeating the same arguments.
– Martin R
Jul 19 at 18:22













@MartinR It was deleted so I think only I can see it but here is the link in case you're able to see deleted Qs math.stackexchange.com/questions/2810441/prove-x-is-bigger-with
– Frank Shmrank
Jul 19 at 18:49




@MartinR It was deleted so I think only I can see it but here is the link in case you're able to see deleted Qs math.stackexchange.com/questions/2810441/prove-x-is-bigger-with
– Frank Shmrank
Jul 19 at 18:49












@Batominovski OK so if we relax the domain/range then can it be answerable?
– Frank Shmrank
Jul 19 at 18:50




@Batominovski OK so if we relax the domain/range then can it be answerable?
– Frank Shmrank
Jul 19 at 18:50















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