William Montgomery

2022-01-26

I want to find the local minima of this equation
${\mathrm{sin}}^{2}\left(\frac{33}{x}\pi \right)+{\mathrm{sin}}^{2}\left(x\pi \right)=y$

Brynn Ortiz

Considering
$y={\mathrm{sin}\left(\frac{33}{x}\pi \right)}^{2}+{\mathrm{sin}\left(x\pi \right)}^{2}$
as said in comments, no roots.
Concerning the extrema, taking derivatives
${y}^{\prime }=2\pi \mathrm{sin}\left(\pi x\right)\mathrm{cos}\left(\pi x\right)-\frac{66\pi \mathrm{sin}\left(\frac{33\pi }{x}\right)\mathrm{cos}\left(\frac{33\pi }{x}\right)}{{x}^{2}}=\pi \left(\mathrm{sin}\left(2\pi x\right)-\frac{33\mathrm{sin}\left(\frac{66\pi }{x}\right)}{{x}^{2}}\right)$
So, assuming $x\ne 0$, the extrema (they are infinitely many) are given the the zero's of the equation
${x}^{2}\mathrm{sin}\left(2\pi x\right)=33\mathrm{sin}\left(\frac{66\pi }{x}\right)$
which is transcendental and then would require numerical methods (remember the equation $x=\mathrm{cos}\left(x\right)$ 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 $\mathrm{sin}\left(2\pi x\right)=0$ which are multiples of half integers.

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