I'm having trouble with a problem regarding finding critical points of a paraboloid defined on a rectangle. The problem is:
Consider the portion of the paraboloid $f(x,y)=16-x^2-y^2$ defined on the rectangle $R=[-1,2] x [-3,4]$.
I'm trying to find any critical points of the paraboloid on the interior of R, relative maxes along each side of R (level curves) where x or y is a constant, and finally identify the global max and mins of f(x,y) on R and where they occur.
I tried with the Hessian determinant but I'm not sure if that's quite right. Thanks
with $$f(x,y)=16-x^2-y^2$$ we get $$\frac{\partial f(x,y)}{\partial x}=-2x$$ and $$\frac{\partial f(x,y)}{\partial y}=-2y$$ and now you must Control the sides of your rectangle.