   Chapter 3.6, Problem 3E

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 6 − 5 x 5 + 25 x 3 − 6 x 2 − 48 x

To determine

To sketch: The graph of a function fx=x6-5x5+25x3-6x2-48x

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=x6-5x5+25x3-6x2-48x

3) Calculation:

fx=x6-5x5+25x3-6x2-48x

By differentiating it,

f'x=6x5-25x4+75x2-12x-48

Solve f'(x)=0

6x5-25x4+75x2-12x-48=0

x=-1.31, x= -0.84, x=1.06  x=2.5, x=2.75

Consider graph of f'(x)

From the graph, f'>0  in the intervals -1.31, -0.84, 1.06, 2.5 and (2.75, )

f'<0 in the interval -, -1.31, -0.84, 1.06 and (2.5, 2.75)

Therefore the function f(x) is increasing in the intervals

-1.31, -0.84, 1

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