   Chapter 3.6, Problem 17E

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 + 5 sin x ,      x ≤ 20

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=x+5sinx x20

3) Calculation:

Graph of f(x)

f'x=1+5cosx2x+5sinx

Graph of f(x)

So from the graph we can say that

Function is increasing in the intervals

-4.91,-4.51, 1.77, 0,4.91, 8.06, 10.79,14.34 and(17.08,20)

Function is decreasing in the interval

-4.51,-4.10, 1.77,4.10, 8.06,10.79, (14.34,17.08)

Function has extreme values

Local max :(-4.51, 0.62),(1.77,2.58),(8.06,3.60),(14.34,4.39)

Local min :(10.79,2.43), (17.08,3.49)

f"x=-1+10cosx+25cos2x+10sinx+50sin2x4(x+5sinx)3

Graph of f(x)

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