Consider the graphs of the functions $f(x)=2\cos\left(\frac{x}{2}\right)+3$ and $g(x)=4$. The number of points of intersection of the two graphs in the interval $[0,4\pi]$ is:

For intersection of the two graphs,

$f(x)=g(x)$

So,

$2\cos\left(\frac{x}{2}\right)+3=4$

$2\cos\left(\frac{x}{2}\right)=1$

$\cos\left(\frac{x}{2}\right)=\frac{1}{2}$

Now,

$x\in [0,4\pi]$

So,

$\frac{x}{2}\in [0,2\pi]$

In the interval $[0,2\pi]$,

$\cos\theta=\frac{1}{2}$ has two solutions:

$\theta=\frac{\pi}{3},\frac{5\pi}{3}$

Here,

$\theta=\frac{x}{2}$

So,

$\frac{x}{2}=\frac{\pi}{3}$ or $\frac{x}{2}=\frac{5\pi}{3}$

Therefore,

$x=\frac{2\pi}{3}$ or $x=\frac{10\pi}{3}$

Hence, the number of points of intersection is $2$.