Chapter 5.3, Problem 90E

a.

To determine

To graph: The given function in the given interval and approximate maximum and minimum points (to four decimal places) of the graph.

Answer to Problem 90E

The approximate maximum and minimum points are:

Maxima: $(3.6652,1.25),(5.7596,1.25)$

Minima: $(1.5708,-1),(4.7124,1)$

Explanation of Solution

Given information:

The following function:

  $f(x)={\cos }^{2}x-\sin x$

The following interval:

  $[0,2\pi )$

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expression ${\cos }^{2}x-\sin x$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

Interpretation:

Using $\boxed{\text{TRACE}}$ feature on the calculator, the maximum and minimum points are obtained as:

Maxima: $(3.6652,1.25),(5.7596,1.25)$

Minima: $(1.5708,-1),(4.7124,1)$

b.

To determine

To solve: The given trigonometric equation and verify that the x-coordinates of the maximum and minimum points of $f$ are among its solutions.

Answer to Problem 90E

The solutionsof the trigonometric equation are:

  $\begin{array}{l}x=\frac{\pi }{2}\text{+2n}\pi \\ x=\frac{3\pi }{2}\text{+2n}\pi \\ x=\frac{7\pi }{6}\text{+2n}\pi \\ x=\frac{11\pi }{6}\text{+2n}\pi \end{array}$

Explanation of Solution

Given information:

The following equation:

  $-2\sin x\cos x-\cos x=0$

Calculation:

The given equation is solved as:

  $\begin{array}{l}-2\sin x\cos x-\cos x=0\text{ [Given] }\\ \cos x(-2\sin x-1)=0\text{ [Factor]}\\ \cos x=0\text{ [Set each factor as 0]}\\ x=\frac{\pi }{2}\text{+2n}\pi \text{,}\frac{3\pi }{2}\text{+2n}\pi \text{ [Using inverse trigonometric function]}\\ \sin x=-\frac{1}{2}\\ x=\frac{7\pi }{6}\text{+2n}\pi ,\frac{11\pi }{6}\text{+2n}\pi \text{ [Using inverse trigonometric function]}\end{array}$

To verify if the x-coordinates of the maximum and minimum points of $f$ are among the solutions of the trigonometric equation, graph both equations in the same window as follows:

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expressions ${\text{Y}}_1={\cos }^{2}x-\sin x$ and ${\text{Y}}_2=-2\sin x\cos x-\cos x$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

Interpretation:

The x-coordinates of the trigonometric function( $-2\sin x\cos x-\cos x$ ) at $0$ is same as the x-coordinates of the maximum and the minimum points of $f(x)$ . Hence, verified.