1. The problem statement, all variables and given/known data Consider a function of one variable, f(x), which is continuous on an interval I. If f has a single critical point, x = a, in I, then if that point is a local maximum (or minimum) it must also be an absolute maximum. Explain why this is. 2. Relevant equations Intermediate Value Theorem Rolle's Theorem 3. The attempt at a solution Critical point x=a, in I, is a local maximum or minimum, thus f'(x)=0 at x=a. The function f thus must either concave up (for a to be a local min) or concave down (for a to be a local max). Since both cases are similar, let consider x=a is a local max. For a local max, f'(x) is decreasing to 0 as x approaches a from the left and f'(x) is increasing negative away from 0 as x deviate away from a on the right. Since there is only one single critical point at x=a, the concavity of the graph will not change. f(x) on the left of is an increasing function to f(a) and f(x) on the right of f is a decreasing function away from f(a). Thus x=a must also be the absolute maximum. Is my reasoning correct? Or did I unconsciously assumed something that I shouldn't have?