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Chapter 1: Q. 49 (page 121)

Use the Extreme Value Theorem to show that each function f has both a maximum and a minimum value on [a, b]. Then use a graphing utility to approximate values M and m in [a, b] at which f has a maximum and a minimum, respectively. You may assume that these functions are continuous everywhere.

f(x)=x4−3x2−2,[a,b]=[−2,2]

Short Answer

Expert verified

M=-2,2m=-1.44,1.44

Step by step solution

01

Step 1. Given information.

We have been given a function and an interval as:

f(x)=x4−3x2−2,[a,b]=[−2,2]

We have to show that this function f has both a maximum and a minimum value on [a, b] using the Extreme Value Theorem.

Also, we have to find approximate values M and m in [a, b] at which f has a maximum and a minimum, respectively, using a graphing utility.

02

Step 2. Apply the Extreme Value Theorem 

limx→−2 f(x)=limx→−2 x4−3x2−2=(−2)4−3(−2)2−2=16−3(4)−2=14-12=2limx→2 f(x)=limx→2 x4−3x2−2=(2)4−3(2)2−2=16−3(4)−2=14-12=2

03

Step 3. Draw the graph of the given function

04

Step 4. Find M and m at which f has a maximum and a minimum 

The maximum value of the function in the interval is M=-2,2.

The maximum value of the function in the interval is m=-1.22,1.22.

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Most popular questions from this chapter

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: For limx→cf(x) to be defined, the function f must be defined at x = c.

(b) True or False: We can calculate a limit of the form limx→cf(x) simply by finding f(c).

(c) True or False: If limx→cf(x)=10, then f(c) = 10.

(d) True or False: If f(c) = 10, then limx→cf(x)=10.

(e) True or False: A function can approach more than one limit as x approaches c.

(f) True or False: If limx→4f(x)=10, then we can make f(x) as close to 4 as we like by choosing values of x sufficiently close to 10.

(g) True or False: If limx→6f(x)=∞, then we can make f(x) as large as we like by choosing values of x sufficiently close to 6.

(h) True or False: If limx→∞f(x)=100, then we can find values of f(x) between 99.9 and 100.1 by choosing values of x that are sufficiently large.

Use the Extreme Value Theorem to show that each function f has both a maximum and a minimum value on [a, b]. Then use a graphing utility to approximate values M and m in [a, b] at which f has a maximum and a minimum, respectively. You may assume that these functions are continuous everywhere.

f(x)=3−2x2+x3,[a,b]=[−1,2]

Describe the punctured interval around x=2that has a radius of 3 and the punctured interval aroundx=4 that has a radius of 0.25.

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) A limit exists if there is some real number that it is equal to.

(b) The limit of fxas x→cis the value fc.

(c) The limit of fxas x→cmight exist even if the value of fcdoes not.

(d) The two-sided limit of fxas x→cexists if and only if the left and right limits of fxexists as x→c.

(e) If the graph of fhas a vertical asymptote at x=5, then limx→5fx=∞.

(f) If limx→5fx=∞, then the graph of fhas a vertical asymptote at x=5.

(g) If limx→2fx=∞, then the graph of fhas a horizontal asymptote at x=2.

(h) Iflimx→∞fx=2, then the graph offhas a horizontal asymptote aty=2.

Sketch a labeled graph of a function that fails to satisfy the hypothesis of the Intermediate Value Theorem, and illustrate on your graph that the conclusion of the Intermediate Value Theorem does not necessarily hold.
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