Chapter 9.4, Problem 49E

To determine

To find: The correct option for the point that determines the maximum value of the graph of the given equation.

Answer to Problem 49E

The correct option is A.

Explanation of Solution

Given:

The given equation $y=-4\cos \left(x-\frac{\pi }{2}\right)$ .

Calculation:

Consider the given function is,

  $y=-4\cos \left(x-\frac{\pi }{2}\right)$

Compare the above equation with the general form $g\left(x\right)=a\cos \left(x-h\right)+k$ .

Then,

  $\begin{array}{l}a=\left|-4\right|\\ b=1\\ h=\frac{\pi }{2}\\ k=0\end{array}$

Consider the maximum value is obtained at the points $\left(\frac{1}{4}\cdot \frac{2\pi }{b},a+k\right)$ .

Then,

  $\begin{array}{c}\left(\frac{1}{4}\cdot \frac{2\pi }{b},a+k\right)=\left(\frac{1}{4}\cdot \frac{2\pi }{1},-4+0\right)\\ =\left(\frac{\pi }{2},-4\right)\end{array}$

As $a<0$ reflect the graph and the graph is reflected in the midline so the maximum value occur at $\left(-\frac{\pi }{2},4\right)$ .

Thus, the correct option is A.