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I have a function $f(x)$, such that at point $x'$, it attains its minimum value, but the gradient at this point $x'$ is not equal to $0$. On the other hand the function $f(x)$ has slightly higher value at point $x''$ but the gradient at this point is equal to $0$.

The function was supposed to be convex

What does this mean?

Carlos Eugenio Thompson Pinzón
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user34790
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  • Disconnected domain and endpoint minimum. Your teacher was in a rather playful mood if he decided to give you this puzzle as a regular homework... – fedja Oct 30 '13 at 22:07
  • @fedja. This is not a homework. Could you please elaborate? I didn't get it – user34790 Oct 30 '13 at 22:51
  • Domain: $[0,1]\cup [4,6]$; $f(x)=x+x^2$ on $[0,1]$, $(x-5)^2+1$ on $[4,6]$; $x'=0$, $x''=5$. If not a homework, what is it? – fedja Oct 30 '13 at 23:29

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