alleryAt a **local maximum** of a differentiable function $f$:
A$f'(x) > 0$
B$f'(x) = 0$ and $f''(x) < 0$
C$f'(x) = 0$ and $f''(x) > 0$
D$f''(x) = 0$
Answer & Solution
Correct answer: B. $f'(x) = 0$ and $f''(x) < 0$
**At local max:** $f'(x) = 0$ (critical/stationary point) AND $f''(x) < 0$ (curve concave down).
**At local min:** $f'(x) = 0$ and $f''(x) > 0$.
$f''(x) = 0$ is inconclusive — use higher-derivative test.
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