$2\pi^2(x-1)^2+4a\cos(2\pi x)-9a^3=0$ For which $a$ has only one solution...

Question

For which values of real parameter $a$ the following equation has only one solution: $$2\pi^2(x-1)^2+4a\cos(2\pi x)-9a^3=0$$

Frankly I have no idea and I hope you'll give me some understandable hint to help me find the way to solve this exercise. Thank you!

Answer 1

Denote $w = 2\pi(x-1)$.

Then $\cos(2\pi x)=\cos(2\pi x - 2\pi) = \cos w$.

Function $$f(w) = w^2+4a\cos w-9a^3$$ is even function.

So, if $w_1$ is solution of equation $$f(w)=0,$$ then $w_2=-w_1$ is solution too.

So, if equation $f(w)=0$ must have only $1$ solution, this solution must be

$$ w=0. $$

Then you'll get equation for $a$:

$$ 4a-9a^3=0. $$

$\cdots$