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Consider the following. (If an answer does not exist, enter DNE.)

[tex]\[ S(x) = x - \cos(x), \quad 0 \leq x \leq 4\pi \][/tex]

(a) Find the interval(s) of increase. (Enter your answer using interval notation.)
[tex]\[
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Find the interval(s) of decrease. (Enter your answer using interval notation.)
[tex]\[
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\][/tex]

(b) Find the local minimum value(s). (Enter your answers as a comma-separated list.)
[tex]\[
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\][/tex]

Find the local maximum value(s). (Enter your answers as a comma-separated list.)
[tex]\[
\square
\][/tex]

(c) Find the inflection points. (Order your answers from smallest to largest [tex]\(x\)[/tex], then from smallest [tex]\(x\)[/tex]-value)

smallest [tex]\(x\)[/tex]-value [tex]\((x, y) = (\square)\)[/tex]

[tex]\[
(x, y) = (\square)
\][/tex]

[tex]\[
(x, y) = (\square)
\][/tex]

largest [tex]\(x\)[/tex]-value [tex]\((x, y) = \square\)[/tex]



Answer :

GinnyAnswer
Let's go through the steps to analyze the function [tex]\( S(x) = x - \cos(x) \)[/tex] over the interval [tex]\( 0 \leq x \leq 4\pi \)[/tex].

### (a) Intervals of Increase and Decrease

To determine where the function is increasing or decreasing, we need to find where its derivative [tex]\( S'(x) \)[/tex] is positive or negative.

1. First Derivative Calculation:
[tex]\[ S'(x) = \frac{d}{dx} \left( x - \cos(x) \right) = 1 + \sin(x) \][/tex]

2. Critical Points:
Find where [tex]\( S'(x) = 0 \)[/tex]:
[tex]\[ 1 + \sin(x) = 0 \implies \sin(x) = -1 \][/tex]
Within the interval [tex]\( 0 \leq x \leq 4\pi \)[/tex], [tex]\(\sin(x) = -1\)[/tex] at:
[tex]\[ x = \frac{3\pi}{2}, \frac{7\pi}{2} \][/tex]

3. Sign of [tex]\( S'(x) \)[/tex] in the Intervals:
- For [tex]\( 0 \leq x < \frac{3\pi}{2} \)[/tex]: [tex]\( 1 + \sin(x) \geq 0 \)[/tex] (since [tex]\(\sin(x) \geq -1\)[/tex])
- For [tex]\( \frac{3\pi}{2} < x < \frac{7\pi}{2} \)[/tex]: [tex]\( 1 + \sin(x) < 0 \)[/tex] (since [tex]\( \sin(x) < -1 \)[/tex])
- For [tex]\( \frac{7\pi}{2} < x \leq 4\pi \)[/tex]: [tex]\( 1 + \sin(x) \geq 0 \)[/tex] (since [tex]\(\sin(x) \geq -1\)[/tex])

4. Intervals:
- Increasing: [tex]\( \left[0, \frac{3\pi}{2}\right) \cup \left(\frac{7\pi}{2}, 4\pi\right] \)[/tex]
- Decreasing: [tex]\( \left(\frac{3\pi}{2}, \frac{7\pi}{2}\right) \)[/tex]

### (b) Local Minima and Maxima

To identify local minima and maxima, we analyze the points where the first derivative changes sign.

1. Critical Points:
- [tex]\( x = \frac{3\pi}{2} \)[/tex]
- [tex]\( S''(x) = \cos(x) \)[/tex]
- [tex]\( S''\left(\frac{3\pi}{2}\right) = \cos\left(\frac{3\pi}{2}\right) = 0 - \text{Indeterminate, check endpoints} \)[/tex]

- [tex]\( x = \frac{7\pi}{2} \)[/tex]
- [tex]\( S''\left(\frac{7\pi}{2}\right) = \cos\left(\frac{7\pi}{2}\right) = 0 - \text{Indeterminate, check endpoints} \)[/tex]

### (c) Inflection Points

To locate inflection points, we find where the second derivative changes sign.

1. Second Derivative Calculation:
[tex]\[ S''(x) = \frac{d}{dx} (1 + \sin(x)) = \cos(x) \][/tex]

2. Inflection Points:
Determine where [tex]\( S''(x) = 0 \)[/tex]:
[tex]\[ \cos(x) = 0 \][/tex]
Within [tex]\( 0 \leq x \leq 4\pi \)[/tex]:
[tex]\[ x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \frac{7\pi}{2} \][/tex]

Check the values of [tex]\( S(x) \)[/tex] at these points:
[tex]\[ S\left(\frac{\pi}{2}\right) = \frac{\pi}{2} - \cos\left(\frac{\pi}{2}\right) = \frac{\pi}{2} \][/tex]
[tex]\[ S\left(\frac{3\pi}{2}\right) = \frac{3\pi}{2} - \cos\left(\frac{3\pi}{2}\right) = \frac{3\pi}{2} \][/tex]
[tex]\[ S\left(\frac{5\pi}{2}\right) = \frac{5\pi}{2} - \cos\left(\frac{5\pi}{2}\right) = \frac{5\pi}{2} \][/tex]
[tex]\[ S\left(\frac{7\pi}{2}\right) = \frac{7\pi}{2} - \cos\left(\frac{7\pi}{2}\right) = \frac{7\pi}{2} \][/tex]

### Summary
(a)
- Intervals of Increase: [tex]\([0, \frac{3\pi}{2}) \cup (\frac{7\pi}{2}, 4\pi]\)[/tex]
- Intervals of Decrease: [tex]\((\frac{3\pi}{2}, \frac{7\pi}{2})\)[/tex]

(b)
- Local Minima: Does not exist in this context.
- Local Maxima: Does not exist in this context.

(c)
- Inflection Points:
- [tex]\( (\frac{\pi}{2}, \frac{\pi}{2}) \)[/tex]
- [tex]\( (\frac{3\pi}{2}, \frac{3\pi}{2}) \)[/tex]
- [tex]\( (\frac{5\pi}{2}, \frac{5\pi}{2}) \)[/tex]
- [tex]\( (\frac{7\pi}{2}, \frac{7\pi}{2}) \)[/tex]

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