Diﬀerential Equation Maximization

Hi, I am learning differential equations, and came across concept of maximization.

If c : [a, b] → R is uniformly bounded and x ∈ C^2 ([a, b]) satisfies the diﬀerential inequality x'' + c(t)x' > 0 on [a, b].

How does this show that the maximum value of x should occur at end–points of the interval?

Can someone please help me on this?

Regards, Salil

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This is the Hopf maximum principle. Treated in many text books about partial differential equations. Being over than 50, I always refer to the book by M. Protter and H. Weinberger, Maximum principles for differential equations. –  Denis Serre Sep 23 '10 at 7:11
ok... maybe the guy from which I got the notes forgot to write that :-) Thanks for pointing me to the correct resource. Now that you have mentioned the principle, I was going through it online, but I am unable to apply it to the specific problem mentioned here. Can you please help me out with that? Thanks again! Salil –  Salil Sep 23 '10 at 7:21
Uhmm, if there is a maximum point in the interior of the interval then it is a local maximum, so $x'=0$. But then $x''\geq 0$ and so the function will achieve a greater value somewhere near. If the function is not constant you get a contradiction. –  Gjergji Zaimi Sep 23 '10 at 7:34
Right. Thanks! Regards, Salil –  Salil Sep 23 '10 at 7:43
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