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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)=3−2x2+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)=3−2x2+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→0 f(x)=limx→0 3−2x2+x3=3−202+03=3−0+0=3limx→2 f(x)=limx→2 3−2x2+x3=3−222+23=3−2(4)+8=3−8+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

For each limit statement in Exercises 41-44, use algebra to find δor Nin terms of εor M, according to the appropriate formal limit definition.

limx→∞x-1x=1, findNin terms ofε.

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

For each limit statement , use algebra to find δ > 0 in terms of ε> 0 so that if 0 < |x − c| < δ, then | f(x) − L| < ε.

limx→21x=12;youmayassumeδ≤1

Write delta-epsilon proofs for each of the limit statements in Exercises 47–60.

limx→2+32x-4=0

For each limit statement , use algebra to find δ > 0 in terms of ε> 0 so that if 0 < |x − c| < δ, then | f(x) − L| < ε.

limx→31x=13;youmayassumeδ≤1
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