Question

how does one determine MULTIPLE min/max points in a periodic function?

Answer 1

That would depend on the function, take for example the function sin(x)
first:
\[\frac{d}{dx}\sin{x}=cos(x)\]
now the min max points occur at
\[cos(x)=0\]
that is at every point on the graph of cos(x) that cos(x)=0 so between the interval
\[[-2\pi,2\pi]\]
you have cos(x)=0 at
\[-\frac{3\pi}{2},-\frac{\pi}{2},\frac{\pi}{2},\frac{3\pi}{2}\]
then take those x values you got from sin(x)'s derivative namely cos(x), and plug them into sin(x) (the function your trying to find the min max points of) and you will find
\[\sin(-\frac{3\pi}{2})=1,\sin(-\frac{\pi}{2})=-1,\sin(\frac{\pi}{2})=1,\sin(\frac{3\pi}{2})=-1\]

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