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I want calculate the maximum and minimum of the following function \begin{equation} f(x)=\Biggl\{ \begin{array}{c} \cos x \ \ \ \ \ x\in(0,\pi] \\ \sin x \ \ \ x\in[-\pi,0] \end{array} \end{equation} The points $x=-\pi/2$, $x=\pi$ are absolute minimum. Instead, the point $x=-\pi$ is a relative maximum. My question is: what happens in $ x = 0 $? It is not an absolute maximum. It can be a relative maximum?

Thank you very much.

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  • $\begingroup$ what is your definition of "relative maximum"? $\endgroup$ – Adam Rubinson Dec 13 '12 at 14:26
  • $\begingroup$ "the largest value that the function takes at a point either within a given neighborhood" $\endgroup$ – Mark Dec 13 '12 at 14:29
  • $\begingroup$ That doesn't make sense $\endgroup$ – Adam Rubinson Dec 13 '12 at 14:31
  • $\begingroup$ Is the definition on Wikipedia. $\endgroup$ – Mark Dec 13 '12 at 14:45
  • $\begingroup$ -1 (as usual) for Wikipedia (the full statement there is poorly written). $\endgroup$ – David Mitra Dec 13 '12 at 14:54
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No, you can draw pictures to see this, or just note that when $\pi>x>0$, $cos x>0$, when $-\pi<x<0$, $sin x<0$, and $f(0)=0.$

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