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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

(a) Find the intervals on which f is increasing or decreasing.

(b) Find the local maximum and minimum values of f.

(c) Find the intervals of concavity and the inflection points.

f(x) = sin x + cos x, 0 ≤ x ≤ 2π

a)

To determine

To find: The intervals on which the function x=π4,x=5π4 on [0,2π] is increasing or decreasing.

Explanation

The given function is, f(x)=sinx+cosx .

Obtain the derivative of f(x) .

f(x)=ddx(sinx+cosx)=cosxsinx

Set f(x)=0 and obtain the critical points.

cosxsinx=0cosx=sinxsinxcosx=1tanx=1

The trigonometry function, tan x = 1 when x=π4,x=5π4 .

Hence, the critical points occurs at x=π4,x=5π4 and in the interval notation (0,π4)(π4,5π4)(5π2,2π) .

Determine the nature of f(x) as follows.

Interval f'(x)=cosxsinx f(x)
0<x<π4 f(π6)=cosπ6sinπ6           =32−<

(b)

To determine

To find: The local maximum and minimum values of the given function f(x)=sinx+cosx on [0,2π] .

(c)

To determine

To find: The intervals of concavity and the inflection points of f(x)=sinx+cosx .

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