   Chapter 3.6, Problem 5E

Chapter
Section
Textbook Problem

# 1–8 Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ' and f " to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f ( x ) = x x 3 + x 2 + 1

To determine

To Produce: The graph of a function

fx=xx3+x2+1

Explanation

1) Concept:

i) Function is increasing if f'x>0  and decreasing if f'x<0 in that particular interval.

ii) If f''x>0 function is concave up and f''x<0 function is concave down in that particular interval.

2) Given:

fx=xx3+x2+1

3) Calculation:

We have,

fx=xx3+x2+1

Vertical asymptote is x=-1.47

Differentiate the function by using quotient rule,

f'(x)=x3+x2+1ddxx-xddxx3+x2+1(x3+x2+1)2

By simplifying,

f'(x)=-2x3-x2+1(x3+x2+1)2

Solve f'x=0

-2x3-x2+1(x3+x2+1)2=0

x=0.66

Graph of f'(x) is

From the graph, f'>0  in the intervals -, -1.47 and (-1.47, 0.66) and f'<0  in the interval (0.66, )

Therefore function is increasing in the intervals -, -1

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