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1. 3.1 Extrema on an Interval Don’t get behind the Do your homework meticulously!!!

2. Definition of Extrema • f(c) is the minimum of f on I if f(c) f(x) x in I. • f(c) is the maximum of f on I if f(c) f(x) x in I. If f is continuous on a closed interval [a,b], then f has both a minimum and a maximum on the interval. If f is defined at c, then c is called a critical number of f if f’(c) = 0 or if f’(c) is undefined at c. Relative Extrema occur only at critical numbers. If f has a relative min or relative max at x = c, then c is a critical number of f.

3. Ex. Find the absolute extrema of f(x) = 3x4 – 4x3 on the interval[-1, 2]. f’(x) = 12x3 – 12x2 0 = 12x3 – 12x2 0 = 12x2(x – 1) x = 0, 1 are the critical numbers Evaluate f at the endpoints of [-1, 2] and the critical #’s. 7 f(-1) = f(0) = f(1) = f(2) = 0 abs. min. -1 16 abs. max.

4. Ex. Find the absolute extrema of f(x) = 2x – 3x2/3 on [-1, 3] = 0 C. N. ‘s are x = 1 because f’(1) = 0 and x = 0 because f’ is undefined Evaluate f at the endpoints of [-1, 3] and the critical #’s. f(-1) = f(0) = f(1) = f(3) = -5 abs. min. 0 abs. max. -1 -.24

5. Ex. Find the absolute extrema of f(x) = 2sin x – cos 2x on . f’(x) = 2cos x + 2sin 2x sin 2x = 2 sin x cos x 0 = 2cos x + 2(2sin x cos x) Factor -1 0 = 2cos x(1 + 2sin x) 3 max 2cos x = 0 cos x = 0 -3 2 min sin x = -1/2 -1 Evaluate f at the endpoints of and the critical #’s. -3 2 min -1