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Chapter 10: Q. 15 (page 777)

A function f that satisfies the hypotheses of Rolle’s Theorem on [−2, 2] and for which there are exactly three values c ∈ (−2, 2) that satisfy the conclusion of the theorem .

Short Answer

Expert verified

Function fx=x3-3x+2satisfied all the conditions of Rolle's theorem .

Step by step solution

01

Step 1. Given information .

Consider the function f that satisfies the hypotheses of Rolle’s Theorem on [−2, 2] and for which there are exactly three values c ∈ (−2, 2) that satisfy the conclusion of the theorem.

02

Step 2. Using Rolle's theorem .

f is continuous on [a, b] and differentiable on (a, b), and if f (a) = f (b) = 0, then there exists at least one value c ∈ (a, b) for which f '(c) = 0.

03

Step 3. Classify Rolle's theorem for function fx=x3-3x+2.

The main conditions of the Mean Value Theorem are:

(a) f(x) must be continuous on [a, b]

(b) f(x) must be differentiable on (a, b).

(c) There is some point c on a,bthat is f'c=0.

Differentiate the function.

fx=x3-3x+2f'x=3x2-3

Put f'c=0.

3c2-3=03c2=3c=±1

As it turns out, there aretwo values of c, and they areboth on our interval!

04

Step 4. Plot the graph .

The graph of the given function is shown below.

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Most popular questions from this chapter

In Exercises 22–29 compute the indicated quantities when u=(2,1,−3),v=(4,0,1),andw=(−2,6,5)

Find the area of the parallelogram determined by the vectors u and v.

In Exercises 36–41 use the given sets of points to find:

(a) A nonzero vector N perpendicular to the plane determined by the points.

(b) Two unit vectors perpendicular to the plane determined by the points.

(c) The area of the triangle determined by the points.

P(1,4,6),Q(−3,5,0),R(3,2,−1)

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The sum formulas in Theorem 4.4 can be applied only to sums whose starting index value is k=1.

(b) True or False: ∑k=0n 1k+1+∑k=1n k2is equal to ∑k=0n k3+k2+1k+1.

(c) True or False: ∑k=1n 1k+1+∑k=0n k2is equal to ∑k=1n k3+k2+1k+1 .

(d) True or False: ∑k=1n 1k+1∑k=1n k2 is equal to ∑k=1n k2k+1.

(e) True or False: ∑k=0m k+∑k=mn kis equal to∑k=0n k.

(f) True or False: ∑k=0n ak=−a0−an+∑k=1n−1 ak.

(g) True or False: ∑k=110 ak2=∑k=110 ak2.

(h) True or False: ∑k=1n ex2=exex+12ex+16.

In Exercises 30–35 compute the indicated quantities when u=(−3,1,−4),v=(2,0,5),andw=(1,3,13)

role="math" localid="1649434688557" v×wandw×v

Use limit rules and the continuity of power functions to prove that every polynomial function is continuous everywhere.
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