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Chapter 3: Q. 83 (page 276)

In Exercises 83–86, use the given derivative f' to find any local extrema and inflection points of f and sketch a possible graph without first finding a formula for f.

f'x=x3-3x2+3x

Short Answer

Expert verified

The possible graph of f is

Step by step solution

01

Step 1. Given Information.

The given derivative function isf'(x)=x3-3x2+3x.

02

Step 2. Apply the first derivative test.

To find the local extrema the first derivative of the function must be zero.

So,

f'(x)=x3-3x2+3x0=x3-3x2+3x0=xx2-3x+3x=0

Thus, by the first derivative test, fhas a local minimum at x=0.The function has no local maxima.

03

Step 3. Finding inflection points.

An inflection point occurs when f''(x)=0.

To find the inflection points, use the second derivative test.

So,

f''(x)=3x2-6x+30=3x2-6x+30=3x2-2x+10=x-13x-3x=1

Thus, the function has an inflection point at x=1. It is positive everywhere. Hence the graph of f will be concave up everywhere.

04

Step 4. Sketch the graph of function f.   

The graph of the function is

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Most popular questions from this chapter

Calculate each of the limits in Exercises 15–20 (a) using

L’Hopital’s rule and (b) without using L’H ˆ opital’s rule.

limx→0cosx-1sinx

Determine whether or not each function f in Exercises 53–60 satisfies 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 ∈ (a, b) that satisfy the conclusion of the Mean Value Theorem.

f(x)=ln(x2+1),[a,b]=[0,1]

Last night at 6 p.m., Linda got up from her blue easy chair. She did not return to her easy chair until she sat down again at 8 p.m. Let s(t) be the distance between Linda and her easy chair t minutes after 6 p.m. last night.

(a) Sketch a possible graph of s(t), and describe what Linda did between 6 p.m. and 8 p.m. according to your graph. (Questions to think about: Will Linda necessarily move in a continuous and differentiable way? What are good ranges for t and s?

(b) Use Rolle’s Theorem to show that at some point between 6 p.m. and 8 p.m., Linda’s velocity v(t) with respect to the easy chair was zero. Find such a place on the graph of s(t).

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)=(x-1)2x+2

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=In2xx
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