Verifying the period of $f(x)=\sin(x)+\cos(x/2)$ It seems clear from the graph of $f(x)=\sin(x)+\cos(x/2)$ that the period $p$ of the function is equal to $4\pi$.

To verify that $4\pi$ is a period of $f(x)$, note that
\begin{align}
\sin(x + 4\pi) + \cos\left(\frac{x + 4\pi}{2}\right) & =\sin(x)\cos(4\pi)+\cos(x)\sin(4\pi)+\cos(x/2)\cos(4\pi/2)-\sin(x/2)\sin(4\pi/2) \\
& =\sin(x)+\cos(x/2)
\end{align}
Thus $4\pi$ is indeed a period of $f$.  My question is, how would one go about trying to prove that $4\pi$ is the smallest $p>0$ such that $f(x+p)=f(x)$?

If $f$ is periodic with period $T$ then so is $f'$. This means that
$$ f'(x) = c \implies f'(x+T) = c.$$
When looking at the graph of $f$ it looks like the solutions of the equation
$$ f'(x) = f'(\pi)$$
are exactly the points $S = \{  \pi + 4k \pi : k \in \mathbb Z\}$ . If we manage to prove this then we are finished since for a smaller period $\tilde T,$ $\pi + \tilde T$ would not be a solution of the equation which contradicts the periodicity of $f'$.
We now solve $f'(x) = f'(\pi).$ By definition of $f$ we have $f'(x) = \cos(x) - \frac 12 \sin \frac x2$ and $f'(\pi) = -1.5$. Let $x = 2u$ and write $-1.5$ as $ -1.5 = -1 - 1/2$ and we have
$$ f'(x) = -1.5 \iff \cos(2u)- \frac 12\sin(u) = -1 - 1/2 \iff \cos(2u)+1 -\frac 1 2 \sin(u)  + 1/2 = 0.$$
Using the identity
$$ \cos(2u) + 1 = 2 -2 \sin^2(u)$$
(which can be deduce from the double angle formula and the $\cos^2 u + \sin^2 u =1)$ we have
$$ -2 \sin^2 u - \frac 1 2 \sin u + 2.5 = 0 \iff -4 \sin^2(u)- \sin(u) + 5 = 0$$
This is a quadratic equation in $\sin(u)$ whose solutions are
$$ \sin(u) = 1, \sin(u) = - 5/4.$$
Since $-5/4 < -1$ we have
$$ f'(x) = -1.5 \iff \sin(x/2) = 1 \iff x = \pi + 4 k \pi, k \in \mathbb Z$$
Therefore $4\pi$ is the smallest possible period of $f$.
