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Understanding Absolute and Relative Extrema in Functions on Closed Intervals

This text elaborates on the definitions and significance of extrema in mathematical functions defined on intervals. It discusses the minimum and maximum values of a function, identifying critical numbers where relative extrema occur. Extrema ensure that if a function is continuous on a closed interval [a, b], both minimum and maximum will exist. The document also provides examples illustrating how to find absolute extrema by evaluating functions at critical points and endpoints. This fundamental topic is crucial for calculus and mathematical analysis.

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Understanding Absolute and Relative Extrema in Functions on Closed Intervals

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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

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