1 extrema notebook. November 25, 2012

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1 Do now as a warm up: Suppose this graph is a function f, defined on [a,b]. What would you say about the value of f at each of these x values: a, x 1, x 2, x 3, x 4, x 5, x 6, and b? What would you say about the value of f ' at these x values: x 1, x 2, x 3, x 4, x 5, and x 6? Nov 16 8:00 AM 11/26/2012 Essential Question How can we use calculus to analyze the graphs of functions? to be able to use derivatives to find extrema of a function Formal definition of extrema: Let f be defined on an interval I containing c. 1) f(c) is the minimum of f on I if f(c) < f(x) for all x in I. 2) f(c) is the maximum of f on I if f(c) > f(x) for all x in I. Nov 18 12:47 PM 1

2 The minimum and maximum of a function on an interval are the extreme values, or extrema, of the function on the interval. The Extreme Value Theorem If f is continuous on a closed interval [a,b], then f has both a minimum and a maximum on the interval. Nov 18 12:54 PM Thm. f is cont's on [a,b] f has a maximum and a minimum f g ex. f is defined on [1,7]. The max is 10 & it occurs at x=1. The min is 2 & it occurs at x=7. ex. g is defined on (1,6). The max is 8 & it occurs at x=3. g has no min. Nov 16 8:01 AM 2

3 The Extreme Value Theorem, like the Intermediate Value Theorem (remember that?) is an existence theorem because it tells of the existence of minimum and maximum values, but does not show how to find these values. Use your calculator to find the maximum and minimum values of these functions on the given interval. 1) f(x) = x 4x + 5 on the closed interval [ 1,3] 2) f(x) = x 2x + x 2 on the closed interval [ 1, 3] Nov 18 12:59 PM Relative extrema If f is a function defined on the open interval (a,b) and c is a point in the interval, then 1) if f(c) is a maximum, it is called a relative maximum of f and 2) if f(c) is a minimum, it is called a relative minimum of f. Informally, you can think of a relative maximum occurring on a "hill" of the graph and a relative minimum occurring in a "valley" of the graph. Nov 18 1:08 PM 3

4 Hills and valleys of graphs can occur in two ways: If the hill or valley is smooth and rounded, then the graph has a horizontal tangent line at the high point or low point. If the hill or valley is sharp and peaked, then the graph represents a function that is not differentiable at the high point or low point. What is the slope of a horizontal tangent line? What is the first derivative of a function at a point at which there is a horizontal tangent line? What is the first derivative of a function at a point where there is a cusp (a point which is not differentiable)? Nov 18 1:15 PM ex. Find the critical values of this function that is defined on all reals. At each critical value, identify whether it is a local or absolute extrema and discuss the value of the derivative. Nov 16 8:03 AM 4

5 Find the value of the derivative at each of the relative extrema in the following graphs. 1) f(x) = x 2) f(x) = sin x Nov 18 1:20 PM Critical Values Let f be defined at c. If f'(c) = 0 or if f is not differentiable at c, then c is a critical value of the function. Relative extrema of functions occur only at critical values of the function. Nov 18 1:30 PM 5

6 ex. Find the critical values of Nov 16 8:04 AM To find the extrema of a continuous function f on a closed interval [a,b] 1. Find the critical values of f in (a,b) 2. Evaluate f at each critical value in (a,b) 3. Evaluate f at each endpoint of [a,b] 4. The least of these values is the minimum. The greatest is the maximum. Example: Find the extrema of f(x) = 3x 4x on the interval [ 1,2]. Nov 18 1:34 PM 6

7 Find the extrema of f(x) = 2x 3x on the interval [ 1,3]. Nov 18 1:40 PM Find the extrema of f(x) = 2 sin x cos 2x on the interval [0,2pi]. Nov 18 1:42 PM 7

8 ex. Find the value of a so that has a local extreme value at x=1. Nov 16 8:05 AM HW p. 169 # 3 8, 13 16, odds Nov 18 1:43 PM 8