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Chapter 3: Q. 22 (page 314)

For each function f that follows, construct sign charts forf, f',and f '', if possible. Examine function values or limits at any interesting values and at 卤鈭�. Then interpret this information to sketch a labeled graph of f.

f(x)=lnx2+1.

Short Answer

Expert verified

Sign charts forf, f',and f ''are the following.

Graph of function is following.

Step by step solution

01

Step 1. Given information. 

The given function isf(x)=lnx2+1.

02

Step 2. Roots of the function. 

Equate the function to zero.

f(x)=0lnx2+1=0x2+1=1x2=0x=0

The root of the function is x=0.

03

Step 3. critical points of the function. 

Determine the first derivative of the function.

f'(x)=ddxlnx2+1=2xx2+1

critical points of the function.

f'(x)=02xx2+1=02x=0x=0

so the function has critical point atx=0.

04

Step 4. inflection point of the function.  

find the second derivative of the function.

f''(x)=ddx2xx2+1=-2x2-1x2+12

the inflection point of the function.

f''(x)=0-2x2-1x2+12=0x2-1=0x2=1x=卤1

So the function has inflection points at x=卤1.

So the sign chart is the following.

05

Step 5. Graph of function. 

The graph of the function is the following.

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Most popular questions from this chapter

Determine whether or not each function fsatisfies the hypotheses of the Mean Value Theorem on the given interval [a,b]. For those that do, use derivatives and algebra to find the exact values of all c鈭�(a,b)that satisfy the conclusion of the Mean Value Theorem.

f(x)=tanx,[a,b]=[-蟺,蟺].

Use a sign chart for f'to determine the intervals on which each function fis increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

role="math" localid="1648370582124" f(x)=sinx.cosx

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=2x-15

Find the possibility graph of its derivative f'.

Use the definitions of increasing and decreasing to argue that f(x)=x4 is decreasing on (鈭�鈭�,0] and increasing on [0,鈭�). Then use derivatives to argue the same thing.
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