`S(x) = x - sin(x), 0<=x<=4pi` (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...

`S(x) = x - sin(x), 0<=x<=4pi` (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.

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Textbook Question

Chapter 4, 4.3 - Problem 44 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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tiburtius | High School Teacher | (Level 2) Educator

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`S(x)=x-sin(x), 0leq x leq 4pi`

(a)

Function is increasing (decreasing) when its derivative is greater (less) than zero. Therefore, we first need to find where the derivative is equal to zero.

`S'(x)=1-cos(x)`

`1-cos(x)=0`

`cos(x)=1`

`x=2k pi, k in ZZ`

Since `0 leq x leq 4pi` and we are not interested in end points, we get

`x=2pi`

Now we know that the derivative has constant sign on these intervals

`(0,2pi)` and `(2pi,4pi)`

Now we simply choose one number from the first interval and check whether the derivative is greater or less than zero and than repeat the process for the second interval.

`S'(\pi)=1-(-1)=2>0`

`S'(3pi)=1-(-1)=2>0`

Therefore, the function is increasing for both of the intervals i.e. the function is increasing on `[0,4pi].`

(b)

Since the function is increasing over the whole domain minimum is achieved in left endpoint and maximum is achieved in right endpoint.

 `S(0)=0`

Minimum is `(0,0).`

`S(4pi)=4pi`

Maximum is `(4pi,4pi).` 

(c)

Function is convex (concave) when its second derivative is greater (less) than zero. So we will basically repeat the same process as in part a) but this time with second derivative.

`S''(x)=sin(x)`

`sin(x)=0`

`x=k pi, k in ZZ`

`x=pi,\ 2pi,\ 3pi`

This means that inflection points are at `pi, 2pi, 3pi.`

Therefore, our intervals are `(0,pi), (pi,2pi), (2pi,3pi)` and `(3pi,4pi).` 

`S(pi/2)=1>0`

`S((3pi)/2)=-1<0`

`S((5pi)/2)=1>0`

`S((7pi)/2)=-1<0`

` `

Function is convex on `(0,pi)` and `(2pi,3pi),` while concave on `(pi,2pi)` and `(3pi,4pi).`

``Part (d) is in the image below.

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