Suppose a function f is continuous on [a,b], differentiable on (a,b), and f′(x)≠0 for all a

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Suppose a function f is continuous on [a,b], differentiable on (a,b), and f′(x)≠0 for all a<x<b. Then f attains both of its absolute extrema at the endpoints a and b.

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To analyze this statement, I’ll examine what the given conditions imply about the behavior of f on [a,b]. First, f is continuous on the closed interval [a,b], and differentiable on the open interval (a,b). By the Extreme Value Theorem, f must attain its absolute maximum and minimum somewhere on [a,b]. Now, inside (a,b) we are told that f'(x) ≠ 0 for all x in (a,b). This eliminates any interior critical points where a local extremum could occur via Fermat’s theorem. Next, recall that a derivative with the Darboux property (the intermediate value property) cannot jump from positive to negative without hitting zero. If f’ were to change sign from positive on some subinterval to negative on another, there would have to be a point where f’(x) = 0, contradicting f'(x) ≠ 0 everywhere in (a,b). Therefore, f’ must have a constant sign on (a,b): either f’(x) > 0 for all x in (a,b) or f’(x) < 0 for all x in (a,b). If f’ > 0 on (a,b), f is strictly increasing, so the absolute minimum occurs at a and the absolute maximum at b. If f’ < 0 on (a,b), f is strictly decreasing, so the absolute minimum occurs at b and the absolute maximum at a. In either case, the absolute extrema occur at the endpoints a and b, not in the interior. Therefore, under the given hypotheses, the statement that f attains both of its absolute extrema at the endpoints a and b is true, and any counterexample would have to violate either continuity on [a,b], differentiability on (a,b), or f'(x) ≠ 0 for all interior x.

MATH_1225_17255_202501 4.1 Minimum and Maximum Values

Suppose a function f is continuous on [a,b], differentiable on (a,b), and f′(x)≠0 for all a<x<b. Then f attains both of its absolute extrema at the endpoints a and b.

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