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In this problem, we would like to find the extreme values of \[f(x,y) = 4-\dfrac{1}{4}(x^2+y^2)\] strictly on the constraint set \[g(x,y) = (x-1)^2 + 4y^2 - 4 = 0.\] At each of the critical points shown in the graph below, the level curve of \(f(x,y)\) (the black curve) is tangent to the constraint set (the blue curve). Since \(\nabla f \) is perpendicular to the level curve of \(f\) and \(\nabla g\) is perpendicular to the constraint set, we have \(\nabla f\) parallel to \(\nabla g\) at the critical points, or simply \[\nabla f = \lambda \nabla g\] for some scalar \(\lambda\).
On the constraint set, the maximum value of \(f\) is \(\dfrac{23}{6} = 3.8333\dots\) and the minimum value of \(f\) is \(\dfrac{7}{4} = 1.75\), both attained when the level curve just touches (tangent to) the constraint set.
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Joseph Lo