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AP CALCULUS NOTES SECTION 5.5 ABSOLUTE MAXIMA AND MINIMA Relative max. and min. are just high and low points in relation to a local interval. They need not be the highest and lowest points on the entire graph. Such points are called absolute max and absolute min. A.) Definition: A function f is said to have an absolute maximum on an interval I at x0 if f x0 is the largest value of f on I. In other words, f x0 f x for all x in the domain of f on the interval I. A function f is said to have an absolute minimum on an interval I at x0 if f x0 is the smallest value of f on I. In other words, f x0 f x for all x in the domain of f on the interval I. If f has either an absolute maximum or an absolute minimum on I at x0 , then f is said to have an absolute extremum on I at x0 . Ex.1.) Use the graph to find x-coordinates of the relative extrema and absolute extrema of f on 0,7 . EXTREME VALUE THEOREM: If f is a continuous function on a finite closed interval a, b , then f has both an absolute maximum and an absolute minimum on a, b . Absolute extremum occur either at the endpoints of the interval or at critical numbers inside the interval. Note: If f has an absolute extremum on an open interval a, b , then it must occur at a critical number of f. Ex.2.) Find the absolute maximum and minimum values of the function 1 f x x 3 3x 2 1; , 4 . The Closed Interval Method: 2 1.) Find the critical #’s on a, b . 2.) Evaluate f at all critical #’s and at the endpoints a and b. 3.) The abs. max. occurs at the largest of these values and the abs. min occurs at the smallest. B.) Absolute Extrema on Infinite Intervals: certain conclusions about the existence of absolute extrema of a continuous function f on , can be drawn from the end behavior. C.) Absolute Extrema on Open Intervals: examine lim f x and lim f x . x a 19.) f x x3 3x 2; , x b 21.) f x x2 ; x 1 5, 1 THEOREM: Suppose that f is continuous and has exactly one relative extremum at x0 on an interval I. 1. If f has a relative minimum at x0 , then f x0 is the absolute minimum of f on I. 2. If f has a relative maximum at x0 , then f x0 is the absolute maximum of f on I. Sample Problems: 3.c.) 4.c.) -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 Ex.3.) Find the absolute maximum and minimum values of f x 10x 2 ln x on the interval 1, e 2 .