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Chapter 3: Q. 36 (page 261)

Use a sign chart for f'to determine the intervals on which each function fis increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

f(x)=cos2(x)

Short Answer

Expert verified

Ans: Increasing interval [2k+1,2k+3]

and decreasing elsewhere.

Step by step solution

01

Step 1. Given information:

f(x)=cos2(x)

02

Step 2. Finding the derivative of the function:

f(x)=cos2(x)f'(x)=2cosx.sinxf'(x)=sin2xletf'(x)=0鈭�sin2x=0鈬�2x=n蟺鈬�x=n蟺2[wherenisanyinteger]鈬�x=2k+1x=2k+1

03

Step 3. Finding increasing and decreasing intervals: 

Intervals of the given function :

f'(x)hasx=2k+2f'(2k+2)=蟺2cos(蟺2(2k+2))=蟺2cos(k+1)蟺=-蟺2whenk=1,3,5,...蟺2whenk=2,4,6,...

f(x)is increasing on the interval2k+1,2k+3

and decreasing elsewhere.

04

Step 4. Verifying algebraic answers with graphs :

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Most popular questions from this chapter

Sketch the graph of a function f with the following properties:

f is continuous and defined on R;

f(0) = 5;

f(鈭�2) = 鈭�3 and f '(鈭�2) = 0;

f '(1) does not exist;

f' is positive only on (鈭�2, 1).

Find the possibility graph of its derivative f'.

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

fx=sexx

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

f(x)=(x-1)2x+2

For each graph of f in Exercises 49鈥�52, explain why f satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b] and approximate any values c 鈭� (a, b) that satisfy the conclusion of the Mean Value Theorem.
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