# 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?

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