My task was to determine the global minimum of a function $f(x,y) = x^3 + y^3 - 3xy$ on the square $[0,2] \times [0,2]$.
I first calculated the points where the gradient $(grad f)(x,y) = (3x^2 - 3y, 3y^2 - 3x)$ becomes the zero vector. That's the case for $(0,0)$ and $(1,1)$. I then took a look at the Hessian matrix in these points, which was indefinite in $(0,0)$ and positive definite in $(1,1)$. I concluded that in $(1,1)$ there is a local minimum and the function has no other turning points.
The sample solution then proceeds to scan the boundary points of the square $[0,2]^2$. However, I think that this is not required. I know that the function has no other turning points (especially no local maximum) that could make the function value drop again, hence the local minimum also has to be the global minimum.
Is my conclusion correct for multidimensional functions? Does it matter whether or not the function of interest is continuous?