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Chapter 3: Q. 9 (page 247)

If a continuous, differentiable function f has values f (−2) = 3 and f (4) = 1, what can you say about f ' on [−2, 4]?

Short Answer

Expert verified

f is continuous and differentiable on -2,4and satisfied all conditions of Rolle's theorem .

Step by step solution

01

Step 1. Given information .

Consider the given points of function f-2,4.

02

Step 2. Find function f to satisfy the given conditions .

fx=x+2x-4has roots x=-2,x=4.

The function that satisfied the conditions of Rolle's theorem is fx=x2-2x-8Since f is a polynomial, it is continuous and differentiable. In particular, it is continuous on -2,4.

Therefore Rolle’s Theorem applies to the function f , and we can conclude that there must exist some value of c ∈ (−2, 4) for which f '(c) = 0. At this value f will have a horizontal tangent line .

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Most popular questions from this chapter

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of f,f',andf'', and examine any relevant limits so that you can describe all key points and behaviors of f.

f(x)=1-x4-2

Prove that the function is increasing on all values of real numbers.

f(x)=x3.

Find the possibility graph of its derivative f'.

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

f(x)=x2(x-1)(x+1)

Explain the difference between two antiderivatives of the function.
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