Question

(a) Graph $f\left(x\right)=-4\cos x\mathrm{and}g\left(x\right)=2\cos x+3$ on the

same Cartesian plane for the interval $\left[0,2\mathrm{\pi }\right].$

(b) Solve $f\left(x\right)=g\left(x\right)$ on the interval $\left[0,2\mathrm{\pi }\right]$and label the points of intersection on the graph drawn in part (a).

(c) Solve $f\left(x\right)>g\left(x\right)$ on the interval $\left[0,2\mathrm{\pi }\right].$

(d) Shade the region bounded by $f\left(x\right)=-4\cos x\mathrm{and}g\left(x\right)=2\cos x+3$between the two points found in part (b) on the graph drawn in part (a).

Short answer

The graph of$f\left(x\right)=-4\cos x\mathrm{and}\mathrm{g}\left(\mathrm{x}\right)=2\mathrm{cosx}+3$on the same cartesian plane for the interval$\left[0,2\mathrm{\pi }\right]$is

Part b. The solution of $f\left(x\right)=g\left(x\right)\mathrm{on}\mathrm{the}\mathrm{interval}\left[0,2\mathrm{\pi }\right]$is $\frac{2\mathrm{\pi }}{3},\frac{4\mathrm{\pi }}{3}$and the points on the graph is

Part c. The solution of$f\left(x\right)>g\left(x\right)\mathrm{on}\mathrm{the}\mathrm{interval}\left[0,2\mathrm{\pi }\right]\mathrm{is}\frac{2\mathrm{\pi }}{3}<\mathrm{x}<\frac{4\mathrm{\pi }}{3}.$

Part d. The shaded region bounded by $f\left(x\right)=-4\cos x\mathrm{and}\mathrm{g}\left(\mathrm{x}\right)=2\mathrm{cosx}+3$between the two points found in part (b) on the graph drawn in part (a) is

Step-by-step solution

Part (a) Step 1. Given Information 

The given functions are$f\left(x\right)=-4\cos x\mathrm{and}\mathrm{g}\left(\mathrm{x}\right)=2\mathrm{cosx}+3$.

We have to graph the given functions on the same cartesian plane for the interval$\left[0,2\mathrm{\pi }\right].$

Part (a)  Step 2. Sketch the graph 

The graph of $f\left(x\right)=-4\cos x\mathrm{and}g\left(x\right)=2\cos x+3$on the same cartesian plane for the interval$\left[0,2\mathrm{\pi }\right]$is

Part (b) Step 1. Solving 

We have to solve $f\left(x\right)=g\left(x\right),0\le x\le 2\mathrm{\pi }$.

$$\begin{gathered} f\left(x\right)=g\left(x\right) \\ -4\cos x=2\cos x+3 \\ -6\cos x=3 \\ \cos x=-\frac{1}{2} \\ x=\frac{2\mathrm{\pi }}{3},\frac{4\mathrm{\pi }}{3} \end{gathered}$$

The points of the intersection on the graph drawn in part (a) is

Part (c) Step 1. Solving 

We have to solve $f\left(x\right)>g\left(x\right)$on the interval $\left[0,2\mathrm{\pi }\right].$

$$\begin{gathered} f\left(x\right)>g\left(x\right) \\ -4\cos x>2\cos x+3 \\ -6\cos x>3 \\ \cos x>-\frac{1}{2} \\ \frac{2\mathrm{\pi }}{3}<x<\frac{4\mathrm{\pi }}{3} \end{gathered}$$

Part (d) Step 1. Shading the region 

The shaded region bounded by $f\left(x\right)=-4\cos x\mathrm{and}\mathrm{g}\left(\mathrm{x}\right)=2\mathrm{cosx}+3$between the two points found in part (b) on the graph drawn in part (a) is