## Absolute maxima and minima of a function on a constraint set (2 variables)

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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.

To view the interactive graph:

1. Make sure you have the latest version of Java 7 installed in your computer. Tablets and smartphones are not supported.