Find the extreme values of f subject to both constraints f(x, y) = 2x^2+3y^2-4x-5, x^2 + y^2 < =16

has only one critical point at . The function has Hessian



which is positive definite for all , which means attains a minimum at the critical point with a value of .

To find the extrema (if any) along the boundary, parameterize it by and , with . On the boundary, we have




Find the critical points along the boundary:




Respectively, plugging these values into gives 11, 47, 43, and 47. We omit the first and third, as we can see the absolute extrema occur when .

Now, solve for for both cases:





so has two absolute maxima at with the same value of 47.