Maxima and Minima of Functions

Transcription

1 Maxima and Minima of Functions Outline of Section 4.2 of Sullivan and Miranda Calculus Sean Ellermeyer Kennesaw State University October 21, 2015 Sean Ellermeyer (Kennesaw State University) Maxima and Minima of Functions October 21, / 13

2 Absolute Extrema Let f be a function whose domain includes some interval I. If there exists some value c in I such that f (c) f (x) for all x in I, then we say that f (c) is the absolute maximum value of f on I and we say that f assumes its absolute maximum value at x = c. Likewise, if there exists some value c in I such that f (c) f (x) for all x in I, then we say that f (c) is the absolute minimum value of f on I and we say that f assumes its absolute minimum value at x = c. Sean Ellermeyer (Kennesaw State University) Maxima and Minima of Functions October 21, / 13

3 Example 1 1) Find the absolute maximum value of the function f (x) = 2x + 4 on the interval I = [ 2, 3]. Also find all points in I at which this absolute maximum value is assumed. 2) Find the absolute minimum value of the function f (x) = 2x + 4 on the interval I = [ 2, 3]. Also find all points in I at which this absolute minimum value is assumed. (Calculus is not needed in order to answer these questions.) Sean Ellermeyer (Kennesaw State University) Maxima and Minima of Functions October 21, / 13

4 Example 2 1) Find the absolute maximum value of the function f (x) = cos (x) on the interval I = [0, 4π]. Also find all points in I at which this absolute maximum value is assumed. 2) Find the absolute minimum value of the function f (x) = cos (x) on the interval I = [0, 4π]. Also find all points in I at which this absolute minimum value is assumed. (Calculus is not needed in order to answer these questions.) Sean Ellermeyer (Kennesaw State University) Maxima and Minima of Functions October 21, / 13

5 Local (Relative) Extrema Let f be a function whose domain includes some interval I. If there exists some open interval J such that J is contained in I and there exist some value c in J such that f (c) f (x) for all x in J, then we say that f (c) is a local (or relative) maximum value of f on I and we say that f assumes this local maximum value at x = c. Likewise, if there exists some open interval J such that J is contained in I and there exist some value c in J such that f (c) f (x) for all x in J, then we say that f (c) is a local (or relative) minimum value of f on I and we say that f assumes this local minimum value at x = c. Sean Ellermeyer (Kennesaw State University) Maxima and Minima of Functions October 21, / 13

6 Example 3 1) Find any local maximum or minimum values of f (x) = 2x + 4 (and where these values are assumed) on the interval I = [ 2, 3]. 2) Find any local maximum or minimum values of f (x) = cos (x) (and where these values are assumed) on the interval I = [0, 2π] Sean Ellermeyer (Kennesaw State University) Maxima and Minima of Functions October 21, / 13

7 The Extreme Value Theorem If the function f is continuous on the closed interval [a, b], then f has both an absolute maximum and an absolute minimum value on [a, b]. Sean Ellermeyer (Kennesaw State University) Maxima and Minima of Functions October 21, / 13

8 Condition That Must be Satisfied for Local Extrema If the function f has a local extremum (maximum or minimum) that is assumed at the point x = c, then either f (c) = 0 or f is not differentiable at c. A number c at which either f (c) = 0 or f is not differentiable at c is called a critical number (or critical point) of f. Thus local extrema can only occur at critical points. Sean Ellermeyer (Kennesaw State University) Maxima and Minima of Functions October 21, / 13

9 Example 4 1) Find all of the critical points of the function f (x) = 2x ) Find all of the critical points of the function f (x) = x 2. 3) Find all of the critical points of the function f (x) = cos (x). 4) Find all of the critical points of the function f (x) = x 2 2 x. Sean Ellermeyer (Kennesaw State University) Maxima and Minima of Functions October 21, / 13

10 Steps for Finding the Absolute Extrema of a Function on a Given Interval Suppose that we want to find the absolute extreme values of some function f on some given closed interval [a, b]. Step 1) Locate all critical numbers of f that lie in the open interval (a, b). Step 2) Evaluate f at each of the critical numbers found in Step 1 and also evaluate f at the endpoints a and b. Step 3) The largest of the values calculated in Step 2 is the absolute maximum value of f on [a, b] and the smallest is the absolute minimum value. Sean Ellermeyer (Kennesaw State University) Maxima and Minima of Functions October 21, / 13

11 Example 5 1) For the function f (x) = 2x + 4 on the interval [ 2, 3], find the absolute extreme values of f and all points at which these extrema are assumed. 2) For the function f (x) = x 2 on the interval [ 2, 3], find the absolute extreme values of f and all points at which these extrema are assumed. 3) For the function f (x) = cos (x) on the interval [0, 4π], find the absolute extreme values of f and all points at which these extrema are assumed. 4) For the function f (x) = x 2 2 x on the interval [ 2, 2], find the absolute extreme values of f and all points at which these extrema are assumed. Sean Ellermeyer (Kennesaw State University) Maxima and Minima of Functions October 21, / 13

12 Example 6: A Maximization Problem Among all possible rectangles that have perimeter 10 feet, find the one whose area is maximum. Sean Ellermeyer (Kennesaw State University) Maxima and Minima of Functions October 21, / 13

13 Homework In Section 4.2, do problems 1 12 (all), (odd), (odd), 71 and 72. Sean Ellermeyer (Kennesaw State University) Maxima and Minima of Functions October 21, / 13