   Chapter 3.6, Problem 7E

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 ) = 6 sin x + cot x ,      - π ≤ x ≤ π

To determine

To Produce:

The graph of

fx=6 sinx+cot x ,    -πxπ

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=6 sinx+cot x ,    -πxπ

Calculation:

fx=6sinx+cot x ,    -πxπ

This function has vertical asymptote at x=0, x=-π and x= π

Differentiate the function with respect to x,

f'x=6cosx-csc2x

Solve f'x=0

6cosx-csc2x =0

x= -1.40, x=-0.44, x=0 , x=0.44, x=1.40

Graph of f'x=6cosx-csc2x  is

From the graph, f'x> 0 in the intervals -1

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