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Let $T\subset\mathbb{R}^2$ be the (closed) triangle bounded by the lines $x+y=4$, $x\ge-1$ and $y\ge-1$. I want to find and classify all the extrema of the function $f(x,y)=-x^2y(x+y-2)$ on the triangle $T$.

I've done the following: Solving $\nabla f=0$ gives the points $(0,0), (0,1)$ and $0,2$. But when I compute the Hessian, I keep getting eigenvalue 0.

Is there someone who can show me how to solve this problem?

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    But according to your function, the critical points are given as $(0,y)$ $(2,0)$ $(1,\frac{1}{2})$. Check it again.2012-06-20

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