Take the derivative, then convert the sum of trig functions to a product and you'll get an explicit solution to this question.
Derivative = 0, after that it's all blank.
Anyway, the functions I built (rudimentary, but I ain't no math prof). are supposed to pick up resonance (for integers, they're basically x = 3n = 5m). I was wondering if the other peaks are also
similar in character, one of 'em occurs at
x=7.5=3⋅2.5=5⋅1.5.
As for the troughs, I haven't examined them as of yet.
Too, was wondering if this is another
modus of understanding/analyzing numbers. How do other number (set as periods of sine) combinations look? Do the waves exhibit different, (easily) identifiable patterns for composite-composite and prime-composite and prime-prime combinations. Incidentally, the example I chose is prime-prime. We could study other classes of numbers too, but of course we'd need to augment my simple trig functions appropriately, customizing them for the task at hand.
The global maxima, patently obvious what they are, can, for example, be used to find LCM. Not offering anything better, just different.
The local max and min look interesting; at the moment all I can say is they too seem to be common multiples (the local maxima), but for
3n=5n=x, n is not an integer. What can be said is that if
3⋅15=5⋅9=45, there'll be a relative max at x = 4.5.
We could, could we?, also use these trig functions to model cyclic phenomena. Maybe I'm being too optimistic. Yes/no/both/neither/stupido!?