   Chapter 3.6, Problem 1E

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 5 − 5 x 4 − x 3 + 28 x 2 − 2 x

To determine

To produce: The graph of a function f(x)=x5-5x4-x3+28x2+2x

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=x5-5x4-x3+28x2-2x

3) Calculation:

Consider the function,

fx=x5-5x4-x3+28x2+2x

By differentiating it,

f'(x)=5x4-20x3-3x2+56x+2

Solve f'x=0

5x4-20x3-3x2+56x+2=0

Solution is, x=-0.04,   x=-1.4

Consider the graph of f'(x)=5x4-20x3-3x2+56x+2

From the graph, f'>0  in the intervals -, -1.4 and (0, )

f'<0 in the interval -1.4, -0.04

Therefore the function f(x) is increasing in the intervals -, -1.4 and (0, ) and decreasing in the interval -1.4, -0.04

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