How to determine local extrema for $f(x) = x\cdot \sin(x) ^ {\sin(x)}$

Question

I need to find the local extrema points of the following function: $f(x) = x\cdot\sin(x) ^ {\sin(x)}$

I was already able to derive to this function: $f'(x) = x (\ln(\sin(x))+1)\cos(x)\sin(x)^{\sin(x)}+\sin(x)^ {\sin(x)}$

Answer 1

The domain of $f$ is where $\sin x>0$ i.e. $$\bigcup_{n\in \Bbb Z}(2n\pi ,2n\pi +\pi)$$ on this domain by equaling the derivative to zero we obtain $$\sin x^{\sin x}=0\\\text{or}\\ x(1+\ln \sin x)\cos x+1=0$$where $\sin x^{\sin x}=0$ is always impossible and the second equation can only be solved numerically. Here is a sketch of the function