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If a relative maximum is the only relative extrema for it's function, and the function is continuous, can it be guaranteed to be the absolute maximum of the function? And a relative minimum which is the only relative extrema would be the absolute minimum? If so, is there any law that states this?

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  • $\begingroup$ Look at the graph of $f(x)=x^2(x-1)$ $\endgroup$ – John Wayland Bales Sep 17 '17 at 23:54
  • $\begingroup$ @JohnWaylandBales did you mean $f(x) = (x^2)(x-1)$? Or $f(x) = x^(2(x-1))$? $\endgroup$ – Jamman00 Sep 18 '17 at 0:10
  • $\begingroup$ The expressions $f(x)=x^2(x-1)$ and $f(x)=(x^2)(x-1)$ are equivalent. $\endgroup$ – John Wayland Bales Sep 18 '17 at 0:13
  • $\begingroup$ There are 2 relative extrema for the function $f(x) = x^2(x-1)$ $\endgroup$ – Jamman00 Sep 18 '17 at 0:33
  • $\begingroup$ Sorry, I misread your question as "If a relative maximum is the only relative maximum $\cdots$". $\endgroup$ – John Wayland Bales Sep 18 '17 at 0:37
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You might want to specify continuity, otherwise there are counter-examples.

$$f(x)=\frac{|x|(x-1)}{x^3}$$

Local, non-global maximum with only one extremum

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