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$f:\mathbf{R} \rightarrow \mathbf{R}$ is twice differentiable. $f''(x) \leq 0$ $\forall x \in \mathbf{R}$. $f$ is also bounded below. Show $f$ is a constant function.

I've got to $$f(x+y)-f(x) \leq f(x)-f(x-y)\, \forall x \in \mathbf{R}, y>0$$ using Rolle's Theorem and I think I'm making an argument about $f'(x)$ being a decreasing function, but I can't see how to get to $f$ constant.

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    $\begingroup$ The intuitive picture: Assume $f(x)>l$, $f'$ is nonincreasing so if it is not $0$ at some point $x_0$, take the tangent to the graph of $f$ there. Since this line has nonzero slope if will cut the lower bound line $y=l$ either on the left or on the right of $x_0$, and the graph of $f$ lies below that tangent, so it will pass below $y=l$ too, therefore $f'$ vanishes on $\mathbb R$. $\endgroup$
    – plm
    May 30 '12 at 11:33
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You know that $f\,'$ is a non-increasing function.

  1. Suppose that $f\,'(x_0)>0$ for some $x_0\in\Bbb R$; show that $\lim\limits_{x\to-\infty}f(x)=-\infty$.

  2. Suppose that $f\,'(x_0)<0$ for some $x_0\in\Bbb R$; show that $\lim\limits_{x\to\infty}f(x)=-\infty$.

Conclude that $f\,'(x)=0$ for all $x\in\Bbb R$.

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  • $\begingroup$ Thanks - that makes much more sense now! $\endgroup$
    – Amy
    May 30 '12 at 11:52
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I'll include a picture and post my comment as an answer. :)

The intuition:

Assume $f(x)>l$, $f'$ is nonincreasing so if it is not $0$ at some point $x_0$, take the tangent to the graph of $f$ there. Since this line has nonzero slope if will cut the lower bound line $y=l$ either on the left or on the right of $x_0$, and the graph of $f$ lies below that tangent (to make this rigorous use the fundamental theorem of calculus), so it will pass below $y=l$ too, therefore $f'$ vanishes on $\mathbb R$.

You can see this situation with the horizontal line $y=-1$ below, for $f(x)=1-x^2$.

Graph of 1-x^2, with 2 tangents.

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We have for $x<y$ that $$f(x)-f(y)=\int_y^xf'(t)dt\leq \int_y^xf'(y)dt=f'(y)(x-y)$$ hence $\inf_t f(t)\leq f(x)\leq f(y)+f'(y)(x-y)$. What happens if $f'(y)\neq 0$?

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  • $\begingroup$ Could you please tell me what really happen if $f'(y)\neq 0$? $\endgroup$
    – Marso
    Jun 7 '12 at 13:20

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