Equation to Find Local Minima of a Sinusoidal Function

Question

I want to find the local minima of this equation $$\sin^2\left(\dfrac{33}{x}\pi\right)+\sin^2(x\pi)=y$$ However, I want to know if there is an equation (or several equations) that describe all local minima (including roots).

Answer 1

Considering $$y=\sin\left(\dfrac{33}{x}\pi\right)^2+\sin(x\pi)^2$$ as said in comments, no roots.

Concerning the extrema, taking derivatives $$y'=2 \pi \sin (\pi x) \cos (\pi x)-\frac{66 \pi \sin \left(\frac{33 \pi }{x}\right) \cos \left(\frac{33 \pi }{x}\right)}{x^2}=\pi \left(\sin (2 \pi x)-\frac{33 \sin \left(\frac{66 \pi }{x}\right)}{x^2}\right)$$ So, assuming $x \neq 0$, the extrema (they are infinitely many) are given the the zero's of the equation $$x^2 \sin (2 \pi x)=33 \sin \left(\frac{66 \pi }{x}\right)$$ which is transcendental and then would require numerical methods (remember the equation $x=\cos(x)$ does not show explicit solutions).

If $x_n$ denotes the solutions, for very large $n$, they will be closer and closer to the solutions of $\sin(2\pi x)=0$ which are multiples of half integers.

Answer 2

Hint: how can you make $y=0$?