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Find the absolute extreme values taken by $f(x,y) = x^2 + 4y^2 + x - 2y$ on the closed region enclosed by the ellipse $1\over4$$x^2 + y^2 = 1$.

I know this might be a basic question but could someone please explain how to solve this problem?

Thanks in advance.

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up vote 1 down vote accepted

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.

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