Try to solve the problem without the constraint first (set the partials to $0$).
If this solution lies on or inside the ellipse, then you have found the minimum (for this $f$). If not, the solution lies on the ellipse and you can solve it in various ways, using Lagrange multipliers, or re-parameterizing the solution space.
Remember that you want to find the extreme values, that is, both the maximum and minimum. Since $f$ is a positive definite form, the minimum may occur inside the ellipse, but the maximum must lie on the ellipse.