We know, with the IVT theorem: if $f(0)=0,f'(0)>0$ and $f(a)<0$ for a continuous function, one can deduce that there exists at least $c \in ]0, a[$ such that $f(c)=0$.could we generalize this result to 2d dimension on a convex domain $\Omega$ using $\nabla f>0$ on the edge of $\Omega$ if we have $f(a) < 0$ for some $a \in \partial\Omega$?
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