Chapter 4.4, Problem 21E

To determine

To Find: The maximum and minimum points and the inflection point. Then graph the viewing rectangle $[-2\pi ,\pi ]by[-1.2,1.2]$

Explanation of Solution

Given:Member of a family function is $f(x)=\sin (x+\sin cx)$ that occur in FM synthesis. The function with $c=3$ start by $f$ in the viewing rectangle $[0,\pi ]by[-1.2,1.2]$ .

Given: $f(x)=\sin (x+\sin cx)$

  

From the graph it is seen that in the range of $[0,\pi ]$ the function appears to have $2$ minima and $2$ maxima points.

  $f\text{'}(x)$ in the range of $[0,2\pi ]$ is the tangent to the $x$ axis at $0.7$

  $f\text{'}\text{'}(x)$ just touch the axis.

In the graph there is a brief change in concavity.

Here we have two maxima at point $(0.59,1),(0.68,1)$ and minimum point is at $(0.64,0.99)$

The inflection point is at the points

  $x=0,0.61,0.66,1.17,2.28,4.01,4.54,5.67,2\pi$

The function is odd

Thus

The function have the rotational symmetry around $((2n+1)\pi ,0)$