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Chapter 3: Q. 9 (page 247)

If a continuous, differentiable function f has values f (鈭�2) = 3 and f (4) = 1, what can you say about f ' on [鈭�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鈥檚 Theorem applies to the function f , and we can conclude that there must exist some value of c 鈭� (鈭�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

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

fx=3x-2x

Sketch the graph of a function f with the following properties:

f is continuous and defined on R;

f(0) = 5;

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Sketch careful, labeled graphs of each function f in Exercises 63鈥�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)=x2-1x2-5x+4

Determine the graph of a function f from the graph of its derivative f'.

Find the possibility graph of its derivative f'.
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