   Chapter 3.6, Problem 16E

Chapter
Section
Textbook Problem

# 15–18 Use a computer algebra system to graph f and to find f ' and f " . Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of f . f ( x ) = x 2 / 3 1 + x + x 4

To determine

To estimate:

The intervals of increase, decrease and extreme values, intervals of concavity and inflection points by using the graph

Explanation

1) Concept:

i) A function f(x) is decreasing if f'x<0  and increasing if f'x>0 for that particular interval.

ii) A function f(x) is concave down if f''x<0  and concave up if f''x>0 for that particular interval.

2) Given:fx=x231+x+x4

2) Calculation:

Graph of f(x)

Function has extreme values

Local max :(-0.72, 1.46)

Local min :(0.61, 0.41)

f'x=x(10x4+x-2)3x43(x4+x+1)2

Graph of f'(x)

So from the graph we can say that

Function is increasing in the intervals -,-0.72 and (0, 0.61)

Function is decreasing in the intervals -0.72, 0 and (0.61, )

f"x=2x2(65x8-14x5-80x4+2x2-8x-1)9x103(x4+x+1)3

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