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Chapter 3: Q. 14 (page 248)

A function that satisfies the hypothesis, and therefore the

conclusion, of the Mean Value Theorem.

Short Answer

Expert verified

At point1,4the function satisfied the conditions of Rolle's theorem .

Step by step solution

01

Step 1. Given information .

A function that satisfies the hypothesis, of the Rolle's theorem .

02

Step 2. ln (x) is indeed continuous on [1,4] and differentiable on (1,4) therefore it satisfies the hypothesis of the mean value theorem. 

The Mean value theorem stated that the secant line connecting the points x1fx1,x2fx2equals the slope of the tangent line at some c in the open interval x1,x2.

The slope of the secant line role="math" localid="1648627193201" m=In4-In14-1=In43.

f'x=1x

Setting the derivative equal to the slope of secant and solving for x .

1x=In43x=3In4

Since this value3In4of x indeed fall in open interval1,4so this satisfied the condition of Mean value theorem .

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Most popular questions from this chapter

For each set of sign charts in Exercises 53鈥�62, sketch a possible graph of f.

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)=x3鈭�2x2+x

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

fx=arctanx.

For each graph of f in Exercises 49鈥�52, explain why f satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b] and approximate any values c 鈭� (a, b) that satisfy the conclusion of the Mean Value Theorem.

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)=x4-x
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