Question

Two students were playing the same game. The difference in points leading score of the game can be modeled by the function Y=5sin x+3 sin x cos x. Within the interval 0

Answer 1

Step 1

Interprete the range of x

The range is 0

This means that x>0 but x<6

Step 2

Differentiating y with respect to x, we have

$\begin{gathered} \frac{dy}{dx}=5\cos x+3(\sin x(-\sin x)+\cos ^2x) \\ \frac{dy}{dx}=5\cos x+3(\cos ^2x-\sin ^2x)=5\cos x+3\cos 2x \\ \frac{d^2y}{d^2x}=-5\sin x-6\sin 2x \end{gathered}$

Step 3: Find the stationary point,

At the stationary points, we have

$\begin{gathered} 5\cos x+3(\cos ^2x-\sin ^2x)=0 \\ 5\cos x+3(2\cos ^2x-1)=0 \\ 6\cos ^2x+5\cos x-3=0 \end{gathered}$$\text{Let m }=\cos x$

Then,

$\begin{gathered} 6m^2+5m-3=0 \\ m=-1.2374,\text{ or }0.4041 \end{gathered}$

Since the values of cos x cannot be less than -1, then the only possibility is

$\begin{gathered} \cos x=0.4041 \\ x=1.1548 \end{gathered}$