To find the maximum of a function on a closed interval it suffices to look for places where the derivative is 0, undefined, or at the boundary of the interval.
Proof, Suppose that x in the middle of the interval has a derivative > 0. Then from the definition of the derivative there exists h > 0 such that x+h is in the interval and 0 < (f(x+h) - f(x))/h. Multiply by h and rearrange to see that f(x+h) > f(x) and therefore x is not where the maximum is achieved. The argument for f'(x) < 0 is similar except that f(x) is exceeded by f(x-h) instead.
Therefore if f achieves a maximum at x, then x is either a boundary point, or a spot where the derivative is 0 or undefined.
Proof, Suppose that x in the middle of the interval has a derivative > 0. Then from the definition of the derivative there exists h > 0 such that x+h is in the interval and 0 < (f(x+h) - f(x))/h. Multiply by h and rearrange to see that f(x+h) > f(x) and therefore x is not where the maximum is achieved. The argument for f'(x) < 0 is similar except that f(x) is exceeded by f(x-h) instead.
Therefore if f achieves a maximum at x, then x is either a boundary point, or a spot where the derivative is 0 or undefined.