I want to find the local minima of this equation \sin^2

William Montgomery

William Montgomery

Answered question

2022-01-26

I want to find the local minima of this equation
sin2(33xπ)+sin2(xπ)=y

Answer & Explanation

Brynn Ortiz

Brynn Ortiz

Beginner2022-01-27Added 12 answers

Considering
y=sin(33xπ)2+sin(xπ)2
as said in comments, no roots.
Concerning the extrema, taking derivatives
y=2πsin(πx)cos(πx)66πsin(33πx)cos(33πx)x2=π(sin(2πx)33sin(66πx)x2)
So, assuming x0, the extrema (they are infinitely many) are given the the zero's of the equation
x2sin(2πx)=33sin(66πx)
which is transcendental and then would require numerical methods (remember the equation x=cos(x) does not show explicit solutions).
If xn denotes the solutions, for very large n, they will be closer and closer to the solutions of sin(2πx)=0 which are multiples of half integers.

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