   Chapter 3.3, Problem 43E

Chapter
Section
Textbook Problem

# 33-44(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts ( a ) − ( c ) to sketch the graph. Check your work with a graphing device if you have one. f ( θ ) = 2 cos θ + cos 2 θ ,   0 ≤ θ ≤ 2 π

To determine

(a)

To find:

The intervals of increase or decrease

Explanation

1) Concept:

Increasing or Decreasing test:

If f'x>0 then f is increasing

If f'x<0 then f is decreasing

2) Given:

fθ=2cosθ+cos2θ  0θ2π

3) Formula:

Sum and difference rule:

ddxfx±gx=ddxfx±ddx(gx)

4) Calculation:

Consider,

fθ=2cosθ+cos2θ  0θ2π

Differentiate with respect to θ,

f'θ=ddθ2cosθ+cos2θ

By using sum and difference rule,

f'θ=ddθ2cosθ+ddθcos2θ

=-2sinθ-2cosθsinθ

= -2

To determine

(b)

To find:

Local maximum and local minimum values

To determine

(c)

To find:

Intervals of concavity and the inflection points

To determine

(d)

To sketch:

The graph by using information from part (a) – (c)

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