Could someone please check this proof for me?

Question: If f(x) is continuous and differentiable in [a,b], show that if for and for , the function is never less than .

Proof:

Note that f is continuous in [a,b] and so it must possess a minimum in the interval. Assume that the minimum for the function f in [a,b] occurs at a point x = c such that . We can suppose that .

By hypothesis . And so . Since then . By the MVT there exists a point C in such that , contradicting the fact that f'(x) must be negative or 0 for all x satisfying . The same proof can be applied if we suppose that . QED