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

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: $$f'(x) = x (\ln(\sin(x))+1)\cos(x)\sin(x)^{\sin(x)}+\sin(x)^ {\sin(x)}$$

• find $x$ where $f'(x)$ is equal to $0$. Then assess the sign change while crossing the roots – Makina Nov 23 '18 at 19:45
• This function is defined in $\bigcup\limits_{n\in\mathbb Z}[2n\pi,2n\pi+\pi]$. – Federico Nov 23 '18 at 19:50
• You first have to take into considerations the extreme points of the intervals – Federico Nov 23 '18 at 19:51
• Then your equation $f'=0$ inside these intervals is highly non-linear and trascendental... – Federico Nov 23 '18 at 19:52
• I don't think much can be said, except numerically – Federico Nov 23 '18 at 19:53

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