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Chapter 3: Q. 41 (page 275)

Use a sign chart for f''to determine the intervals on which each function f in Exercises 41鈥�52 is concave up or concave down, and identify the locations of any inflection points. Then verify your algebraic answers with graphs from a calculator or graphing utility

f(x)=(x-2)4

Short Answer

Expert verified

The function has no inflection points and is concave up on both(-鈭�,2)and(2,鈭�).

Step by step solution

01

Step 1. Given Information.

The given function isf(x)=(x-2)4.

02

Step 2. Second derivative.

On differentiating the given function, we get,

f'(x)=ddx(x-2)4=4(x-2)4-1ddx(x-2)=4(x-2)3f''(x)=ddx4(x-2)3=12(x-2)2

03

Step 3.Sign Chart.

Now,

f''(x)=0whenx=2,

Therefore, the sign chart will be,

Since, the sign does not change there is no inflection point.

And the function is concave up in(-鈭�,2)and(2,鈭�).

04

Step 4.Verification.

The graph of the function is ,

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Most popular questions from this chapter

Restate Theorem 3.3 so that its conclusion has to do with

tangent lines.

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)=1x2+3

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)=ln(x2+1)

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

f(x)=x3.
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