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Local extrema occur at critical points where the derivative changes sign. A negative second derivative means the curve bends downward, giving a local maximum; a positive one gives a local minimum.
Second derivative test
f'(c) = 0 and f''(c) < 0 ⇒ local maximum; f''(c) > 0 ⇒ local minimum
f' = 3x² − 12 gives x = ±2, with a maximum at x = −2 (f = 16) and a minimum at x = 2 (f = −16).
No. A cubic has local extrema but is unbounded, so neither is a global maximum or minimum.