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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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what is your definition of "relative maximum"? –  Adam Rubinson Dec 13 '12 at 14:26
    
"the largest value that the function takes at a point either within a given neighborhood" –  Mark Dec 13 '12 at 14:29
    
That doesn't make sense –  Adam Rubinson Dec 13 '12 at 14:31
    
Is the definition on Wikipedia. –  Mark Dec 13 '12 at 14:45
    
-1 (as usual) for Wikipedia (the full statement there is poorly written). –  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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