trigonometry - Maxima and minima of $operatorname{sinc}$ function

The function $\operatorname{sinc}{\pi x}$ has maxima and minima given by the function's intersections with $\cos \pi x$, or alternatively by $\frac {d}{dx}\operatorname{sinc}{\pi x}=0$.

Mathematica tells me that

$$\frac {d}{dx}\operatorname{sinc}{\pi x}=\pi \Bigl(\frac {\cos \pi x}{\pi x}-\frac {\sin \pi x}{\pi^2 x^2}\Bigr)$$

So question 1, how do I prove this?

And question 2, how do I derive an equation for all maxima and minima?

Answer

Set derivative equal to 0.

You will after some manipulation like multiplying $(\pi x)^2$ and dividing by $\pi$ get $$\pi x \cos(\pi x) = \sin(\pi x)$$ and equivalently by dividing both sides by $\pi x$

$$\cos(\pi x) = \frac{\sin(\pi x)}{\pi x}$$
Now the right hand side is $\text{sinc}(\pi x)$ and left hand side is the function you want to show it should intersect.

So we are done showing where the extrema are.

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Now to show which are max and which are min.

Sinc as a function is a multiplication between $\frac{1}{\pi x}$ and $\sin(\pi x)$

On $\mathbb R^+$ the first of these is monotonically decreasing and positive. Sin is periodic and alternating +1 -1. Both functions are continuous. We can now use an argument with theorem of intermediate value to show it will be alternatingly max and min with as many maxes as mins.