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This question was asked in an entrance test for an undergraduate mathematics program today, held all over India.

Question: $f$ is a differentiable function in $[0,1]$ such that $f(f(x))=x$ and $f(0)=1$.

Find the value of $\int_{0}^{1} (x-f(x))^{2016} dx$.

I tried to solve it by substituting $f(f(x))$ in place of $x$, but could not proceed much further. Any hints or solutions will be appreciated.

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    $\begingroup$ Hint: the obvious function satisfying the conditions is $1-x$. $\endgroup$ – almagest May 8 '16 at 14:19
  • $\begingroup$ Which test was this asked in? $\endgroup$ – Nikunj May 8 '16 at 14:32
  • $\begingroup$ @Nikunj For ISI - Indian Statistics Institute. $\endgroup$ – Gummy bears May 8 '16 at 15:52
  • $\begingroup$ @almagest Is this operation valid for reaching that conclusion? If we apply Lagrange's MVT on $f$ which is differentiable in $(0,1)$ & continuous on $[0,1]$, $\exists$ at least one $c$ $\in (0,1)$ such that $f'(c)=\frac{f(1)-f(0)}{1-0}=-1$ [since $f(1)=0$ ]. So one possible function is $f(x)=-x+p.$ Now $p=1$ for $x=0,$ hence $f(x)=1-x.$ $\endgroup$ – StubbornAtom Jun 30 '16 at 9:45
  • $\begingroup$ @StubbornAtom we can further assume that $c=1/2$, because of the symmetry. Infinite such functions are possible, so you can't really say anything based on that alone $\endgroup$ – Aditya De Saha Jun 30 '16 at 11:15
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Let

$$ I = \int_0^1 (x - f(x))^{2016}\, dx $$

Substitute $ x = f(u) $ and note that $ f(0) = 1 $, $ f(1) = 0 $ to obtain

$$ I = \int_1^0 (f(u) - u)^{2016} f'(u)\, du = - \int_0^1 (x - f(x))^{2016} f'(x)\, dx $$

Then,

$$ 2I = I + I = \int_{0}^{1} (x - f(x))^{2016} (1 - f'(x))\, dx $$

and we may substitute $ w = x - f(x) $, $ dw = (1 - f'(x))\, dx $ (noting that $1 - f(1) = 1 $ and $ 0 - f(0) = -1 $) to obtain

$$ 2I = \int_{-1}^{1} w^{2016}\, dw = \frac{2}{2017} $$

and

$$ I = \int_0^1 (x - f(x))^{2016}\, dx = \frac{1}{2017} $$

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    $\begingroup$ Why are you so quick at typing? I had been typing this. Well, doesn't matter. +1. $\endgroup$ – S.C.B. May 8 '16 at 14:23
  • $\begingroup$ It must be my lucky day today :) $\endgroup$ – Starfall May 8 '16 at 14:27
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    $\begingroup$ So we need not know what $f(x)$ is in this solution. But I was wondering if for every real number $k>0$, $f(x) = \displaystyle\left(1-x^{\frac{1}{k}}\right)^k$ satisfies all conditions in the problem. $\endgroup$ – StubbornAtom Jun 30 '16 at 9:42
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An alternative solution is to note that $f(f(x))=x$ means that $f(x)$ is its own inverse. Geometrically, this means that the function will be perpendicular to the line $y=x$ at the point of intersection and symmetric on each side of $y=x$. Since it is differentiable over $[0,1]$ and we are given that $f(0)=1$, one such function that comes to mind is $f(x)=1-x$.

Now that a possible function is known, the calculation of the integral is easy:

$$\int_0^1 (x-f(x))^{2016}\,dx=\int_0^1(2x-1)^{2016}\,dx=\frac{1}{2017}$$

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    $\begingroup$ Yes, but you have to show that the value of the integral is the same for all such functions, which makes this solution invalid. (Unless you are in a multiple choice test, then it is clearly the superior solution :)) $\endgroup$ – Starfall May 8 '16 at 14:36
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    $\begingroup$ That is correct, my solution does not show that $\frac{1}{2017}$ is the value of the integral in all such cases. It is purely here to give an explicit example for the OP if he is interested in such an example. $\endgroup$ – Ethan Hunt May 8 '16 at 14:40
  • $\begingroup$ I was so close to solving this question.... I kept on trying with $x-1$ and thinking it seems right... but won't satisfy the condition. facepalm from my side. $\endgroup$ – Gummy bears May 8 '16 at 15:54
  • $\begingroup$ @Starfall Unfortunately this was not a multiple choice question. $\endgroup$ – Gummy bears May 8 '16 at 15:54

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