Local Extrema, Lecture notes of Calculus

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Local Extrema

Definition Let S be the domain of f such that c is an element of S. Then, 1 ) f(c) is a local maximum value of f if there exists an interval (a,b) containing c such that f(c) is the maximum value of f on (a,b)∩S. 2 ) f(c) is a local minimum value of f if there exists an interval (a,b) containing c such that f(c) is the minimum value of f on (a,b)∩S. 3 ) f(c) is a local extreme value of f if it is either a local maximum or local minimum value.

EX 1 Determine local maximum and minimum points for. EX 2 Find all local maximum and minimum points for.

Theorem: Second Derivative Test Let f' and f'' exist at every point on the interval (a,b) containing c and f'(c) = 0. 1 ) If f''(c) < 0 , then f(c) is a local maximum. 2 ) If f''(c) > 0 , the f(c) is a local minimum. EX 3 Find all critical points for. .

EX 5 Let f be continuous such that f' has the following graph. Try to sketch a graph of f(x) and answer these questions. a) Where is f increasing? b) Where is f decreasing? c) Where is f concave up? d) Where is f concave down? e) Where are inflections points? f) Where are local max/min values?

f'(x)

• 3 - 2 - 1 1 2 3 x x