   Chapter 3.6, Problem 6E

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 − x 2 ,    -5 ≤ x ≤ 3

To determine

To Produce: The graph of a function

fx=6sinx-x2 , -5x3

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=6sinx-x2 , -5x3

3) Calculation:

fx=6sinx-x2 , -5x3

Differentiate with respect to x,

f'x=6 cosx-2x ,

Solving f'x=0

6 cosx-2x =0

x= -2.94, x= -2.66, x=1.17

Graph of f'x=6 cosx-2x is

From the graph,

f'x>0 in the interval -5, -2

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