Which condition is required for the Extreme Value Theorem to apply?

Which condition is required for the Extreme Value Theorem to apply?

• A. It must be differentiable on (a,b).
• B. It must be monotonic on [a,b].
• C. It must be continuous on [a,b].
• D. It must be integrable on [a,b].

Answer: (C)

A function that is continuous on a closed interval [a, b] must reach its largest and smallest values somewhere in that interval. This is the Extreme Value Theorem. The idea rests on the interval being compact and the function being continuous: continuity prevents jumps that could skip over potential extrema, and the closed, bounded interval guarantees the function’s values don’t go off to infinity and actually attain extreme values. So there exist points in [a, b] where the function attains its maximum and its minimum. Differentiability on the interior, monotonicity, or integrability alone do not guarantee extremes in the same way. A function can be continuous and still not be differentiable at some points, or not monotone, yet still have maxima and minima on the interval. Similarly, integrability by itself does not ensure the existence of maximum or minimum values.

A function that is continuous on a closed interval [a, b] must reach its largest and smallest values somewhere in that interval. This is the Extreme Value Theorem. The idea rests on the interval being compact and the function being continuous: continuity prevents jumps that could skip over potential extrema, and the closed, bounded interval guarantees the function’s values don’t go off to infinity and actually attain extreme values. So there exist points in [a, b] where the function attains its maximum and its minimum.

Differentiability on the interior, monotonicity, or integrability alone do not guarantee extremes in the same way. A function can be continuous and still not be differentiable at some points, or not monotone, yet still have maxima and minima on the interval. Similarly, integrability by itself does not ensure the existence of maximum or minimum values.