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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鈭�3x2鈭�2,[a,b]=[鈭�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鈭�3x2鈭�2,[a,b]=[鈭�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鈫�鈭�2鈥�f(x)=limx鈫�鈭�2鈥�x4鈭�3x2鈭�2=(鈭�2)4鈭�3(鈭�2)2鈭�2=16鈭�3(4)鈭�2=14-12=2limx鈫�2鈥�f(x)=limx鈫�2鈥�x4鈭�3x2鈭�2=(2)4鈭�3(2)2鈭�2=16鈭�3(4)鈭�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

Calculate each of the limits:

limh鈫�0(2+h)2-22h.

limx鈫�0sinxcosx

For each limit statement , use algebra to find 未 > 0 in terms of 蔚> 0 so that if 0 < |x 鈭� c| < 未, then | f(x) 鈭� L| < 蔚.

limx鈫�-2(4-2x)=8

Write each of the inequalities in interval notation:

x2-1+3<0.5

Sketch a labeled graph of a function that satisfies the hypothesis of the Extreme Value Theorem, and illustrate on your graph that the conclusion of the Extreme Value Theorem follows.
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