The Extreme Value Theorem

Real Analysis · Axiom Academy

LESSON The Extreme Value Theorem A function continuous on a closed, bounded interval cannot dodge its own extremes — it must actually reach both a highest and a lowest value. Take any continuous f on a closed, bounded interval [a,b] — an unbroken curve over a segment that includes its endpoints. The theorem promises two special points: an where f is largest and an where f is smallest, with every other value of f trapped between them. Let f(x) = -(x-1)^2 + 4 on the closed interval . It's a polynomial, so continuous everywhere; the interval is closed and bounded, so the theorem applies. To find the extrema, check the one interior critical point ( f'(x) = -2(x-1) = 0 at x=1 ) against the two endpoints. critical point at x=1 , then compare with the endpoints x=-1 and x=4 The maximum lands in the interior , the minimum at an endpoint . That's exactly why optimizing a continuous function on [a,b] means checking critical points and endpoints: the theorem guarantees the winner is somewhere in that finite list. 3. Drop "Closed": the Extremes Escape Keep the function continuous but open the interval. On (0,1) , the function f(x)=x climbs toward 1 and sinks toward 0 — yet 0 and 1 have been excluded. For any candidate x in the interval, the midpoint is also inside and strictly larger, so no point can be the maximum. and , so the values are bounded. But and , so neither bound is a value of f . No largest and no smallest element exist: and both fail to be attained.

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