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Chapter 1: Q. 53 (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)=3鈭�2x2+x3,[a,b]=[0,2]

Short Answer

Expert verified

M=0,2m=1.33

Step by step solution

01

Step 1. Given information.

We have been given a function and an interval as:

f(x)=3鈭�2x2+x3,[a,b]=[0,2]

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�0鈥�f(x)=limx鈫�0鈥�3鈭�2x2+x3=3鈭�202+03=3鈭�0+0=3limx鈫�2鈥�f(x)=limx鈫�2鈥�3鈭�2x2+x3=3鈭�222+23=3鈭�2(4)+8=3鈭�8+8=3

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=0,2.

The value of the function is minimum in the interval at x=1.33.

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Most popular questions from this chapter

Write delta-epsilon proofs for each of the limit statements limx鈫�cfx=Lin Exercises 47鈥�60.

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