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Chapter 1: Q. 51 (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�1,1]

Short Answer

Expert verified

M=0m=-1,1

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�1,1]

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�1鈥�f(x)=limx鈫�鈭�1鈥�x4鈭�3x2鈭�2=(鈭�1)4鈭�3(鈭�1)2鈭�2=1鈭�3(1)鈭�2=鈭�1鈭�3=-4limx鈫�1鈥�f(x)=limx鈫�1鈥�x4鈭�3x2鈭�2=(1)4鈭�3(1)2鈭�2=1鈭�3(1)鈭�2=鈭�1鈭�3=-4

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 value of the function is maximum in the interval at x=0.

The value of the function is minimum in the interval at x=-1,1.

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Most popular questions from this chapter

Explain why the Intermediate Value Theorem allows us to say that a function can change sign only at discontinuities and zeroes.

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limx鈫�-1x.

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