# How to find a function given its local maximum and minimum points

I know how to find the local maximum and minimum points of a function. Is there a way to reverse the process, so we can find a function ( or even an infinite set of functions ) based on its given local maximum and minimum points ?.

I'm particularly interested in trigonometric functions, because I'd like to know how to find a function to match a given graph of a sound wave ( with constant frequency ). I'm aware that this is related to Fourier transform, but I don't know specifically which calculations are relevant to my question, or which fields of math I need to learn ( my current math knowledge is basically at high school level ).

• For example, I've entered the function $y = \sin\left(x\right) + \sin\left(2x\right)$ into a graphing calculator.
• I see that in the interval $0 \leq x < 2\pi$ there is a maximum point $\left(0.936, 1.76\right)$, a minimum point $\left(2.574, -0.369\right)$, a maximum point $\left(3.709, 0.369\right)$ and a minimum point $\left(5.347, -1.76\right)$.

Given that information, can we find a trigonometric function that matches that data, or at least closely resembles it ?.

I've been searching on this site as well as on Google before posting this question, but haven't found my answer yet. Any help is greatly appreciated !.

• Of course there are infinitely many functions that have max and min at some finite set of points... It is pretty easy to construct a polynomial function that will interpolate through as many points as required, with max and min where required. I don't know of a way to this easily with a Fourier series or similar. – Doug M Aug 3 '17 at 19:29
• – Felix Marin Aug 3 '17 at 19:49