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Chapter 1: Q. 19 (page 120)

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.

Short Answer

Expert verified

A labeled graph of a function that fails to satisfy the hypothesis of the Extreme Value Theorem is

and

Step by step solution

01

Step 1. Given Information. 

The Extreme Value Theorem states that iff is continuous on a closed interval[a, b], then there exist valuesM andm in the interval[a, b]such thatf(M) is the maximum value off(x) on[a, b] andf(m) is the minimum value off(x) on[a, b].

02

Step 2. Sketching a graph 

In the following graph, there is no minimum value on the interval [a, b].

In the following graph, there is no maximum value on the interval[a,b].

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Most popular questions from this chapter

Find a formula for the cost Crof producing a gourmet soup can with radius rand height 5inches, and answer the following questions:

  1. What is the radius of a can that is 5inches tall and costs 30cents to produce?
  2. Your manager wants you to produce 5-inch-tall cans that cost between 20and 40cents. Write this requirement as an absolute value inequality.
  3. What range of radii would satisfy your manager? Write an absolute value inequality whose solution set lies inside this range of radii.

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

limx→0sinxcosx

Calculate each of the limits:

limx→π2cotxcosx.

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.

limx→2-f(x)=2,limx→2+f(x)=1,f(2)=1.
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