Question

(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.

You may want to check your work with a graphing calculator or computer.

58. \(S\left( x \right) = x - \sin x,{\rm{ 0}} \le x \le 4\pi \)

Short answer

1. Function \(f\) is increasing on \(\left( {0,4\pi } \right)\).
2. There is no local minimum or maximum values.
3. Intervals of concave upward: \(\left( {0,\pi } \right)\) and \(\left( {2\pi ,3\pi } \right)\)

Intervals of concave downward: \(\left( {\pi ,2\pi } \right)\) and \(\left( {3\pi ,4\pi } \right)\)Inflection points: \(\left( {\pi ,\pi } \right)\), \(\left( {2\pi ,2\pi } \right)\), and \(\left( {3\pi ,3\pi } \right)\)

d.

Step-by-step solution

Increasing/Decreasing Test

9. In an interval,\(f\)is increasing if \(f'\left( x \right) > 0\) on that interval.
10. In an interval, \(f\) is decreasing if \(f'\left( x \right) < 0\) on that interval.

Find the interval for increase or decrease

(a). The given function \(S\left( x \right) = x - \sin x\).

Find the derivative of \(S\left( x \right) = x - \sin x\) with respect to \(x\).

\(\begin{array}{c}S'\left( x \right) = \frac{d}{{dx}}\left( {x - \sin x} \right)\\ = 1 - \cos x\end{array}\)

Moreover, when \(S'\left( x \right) = 0\), \(\theta = 0,2\pi ,4\pi \).

Draw a table for the interval of increasing and decreasing.

Interval	\(f'\left( x \right)\)	Behavior of \(f\)
\(0 < x < 2\pi \)	+	Increasing on \(\left( {0,2\pi } \right)\)
\(2\pi < x < 4\pi \)	+	Increasing on \(\left( {2\pi ,4\pi } \right)\)

The first derivative test

Assume a critical point \(c\) of a continuous function \(f\).

9. The function \(f\) has a local maximum at the point \(c\) if \(f'\) changes from positive to negative.
10. The function \(f\) has a local minimum at the point \(c\) if \(f'\) changes from negative to positive.
11. The function \(f\) does not have any local maximum or minimum at \(c\) if \(f'\) is positive/negative to the left and right of \(c\).

Find local maximum and minimum values

(b) From the table in Step 2, it can be observed that \(f\) is positive on both sides of the point, so there is no local maximum or minimum.

Concavity test

Concave upward on \(I\): if \(f''\left( x \right) > 0\) on interval \(I\)

Concave downward on \(I\): if \(f''\left( x \right) < 0\) on interval \(I\)

Inflection point

A point \(P\) is known as an inflection point if the function \(f\) is continuous and the curve changes from concave upward to concave downward or concave downward to concave upward at the point \(P\).

Find the interval of concavity and inflection point

(c) Find the double derivative of the function \(S\left( x \right) = x - \sin x\).

\(\begin{array}{c}S''\left( \theta \right) = \frac{d}{{dx}}\left( {1 - \cos x} \right)\\ = \sin x\end{array}\)

For \(S''\left( x \right) = 0\), solve for \(x = 0,\pi ,2\pi ,3\pi ,4\pi \) when \(\left( {0,4\pi } \right)\).

The given function is not continuous at \(x = 0,4\pi \); therefore, \(x = \pi ,2\pi ,3\pi \).

Draw a table for concavity for different intervals.

Interval	Sign of \(f''\left( \theta \right)\)	Behavior of \(f\)
\(\left( {0,\pi } \right)\)	+	Concave upward
\(\theta = \pi \)	0	Inflection
\(\left( {\pi ,2\pi } \right)\)	-	Concave downward
\(\theta = 2\pi \)	0	Inflection
\(\left( {2\pi ,3\pi } \right)\)	+	Concave upward
\(\theta = 3\pi \)	0	Inflection point
\(\left( {3\pi ,4\pi } \right)\)	-	Concave downward

Find \(S\left( \pi \right)\), \(S\left( {2\pi } \right)\), and \(S\left( {3\pi } \right)\) as from the table, you will get the inflection point.

\(S\left( \pi \right) = \pi \)

\(S\left( {2\pi } \right) = 2\pi \)

\(S\left( {3\pi } \right) = 3\pi \)

So, the inflection points are \(\left( {\pi ,\pi } \right)\), \(\left( {2\pi ,2\pi } \right)\), and \(\left( {3\pi ,3\pi } \right)\).

Graph of \(S\)

(d)

Draw the graph of the given functions by using the obtained information.