Chapter 5.3, Problem 93E

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 93E

The approximate maximum and minimum points are:

Maxima: $(1.0472,1.2500),(5.236,1.2500)$

Minima: $(0,1),(3.1416,-1),(6.2832,1)$

Explanation of Solution

Given information:

The following function:

  $f(x)={\sin }^{2}x+\cos x$

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 ${\sin }^{2}x+\cos x$ 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: $(1.0472,1.2500),(5.236,1.2500)$

Minima: $(0,1),(3.1416,-1),(6.2832,1)$

b.

To determine

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

Answer to Problem 93E

The solutions of the trigonometric equation are:

  $\begin{array}{l}x=n\pi \\ x=\frac{\pi }{3}+2n\pi \\ x=\frac{5\pi }{3}\text{+2n}\pi \end{array}$

Explanation of Solution

Given information:

The following equation:

  $2\sin x\cos x-\sin x=0$

Calculation:

The given equation is solved as:

  $\begin{array}{l}2\sin x\cos x-\sin x=0\text{ [Given] }\\ \sin x(2\cos x-1)=0\text{ [Factor] }\\ \sin x=0\text{ [Set each factor to 0] }\\ x=n\pi \text{ [Using inverse trigonometric function]}\\ 2\cos x=1\\ \cos x=\frac{1}{2}\text{ [Divide both sides by 2]}\\ x=\frac{\pi }{3}+2n\pi ,\frac{5\pi }{3}\text{+2n}\pi \text{ [Using inverse trigonometric function]}\end{array}$

On substituting the x-coordinates of the maximum and minimum points of $f$ , the following solutions are obtained:

  $\begin{array}{l}2\sin x\cos x-\sin x=2\sin 0\cos 0-\sin 0=0\\ 2\sin x\cos x-\sin x=2\sin (1.0472)\cos (1.0472)-\sin (1.0472)=0\\ 2\sin x\cos x-\sin x=2\sin (5.236)\cos (5.236)-\sin (5.236)=0\\ 2\sin x\cos x-\sin x=2\sin (3.1416)\cos (3.1416)-\sin (3.1416)=0\\ 2\sin x\cos x-\sin x=2\sin (6.2832)\cos (6.2832)-\sin (6.2832)=0\end{array}$

Since every answer corresponds to $0$ , the x-coordinates of the maximum and minimum points of $f$are among the solutions.