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

Beginner2022-01-27Added 12 answers

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

${y}^{\prime}=2\pi \mathrm{sin}\left(\pi x\right)\mathrm{cos}\left(\pi x\right)-\frac{66\pi \mathrm{sin}\left(\frac{33\pi}{x}\right)\mathrm{cos}\left(\frac{33\pi}{x}\right)}{{x}^{2}}=\pi (\mathrm{sin}\left(2\pi x\right)-\frac{33\mathrm{sin}\left(\frac{66\pi}{x}\right)}{{x}^{2}})$

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.

as said in comments, no roots.

Concerning the extrema, taking derivatives

So, assuming

which is transcendental and then would require numerical methods (remember the equation

If

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