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

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

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