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This was a question on my math exam. I received zero points, so I obviously don't know how to do it. How would I go about this:

Find the shortest distance from the point $(1, 1, 1)$ to the surface $z = xy$.

I don't know how to go about this. Please help me!

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I'm somewhat concerned that the solution might not be covered in our material, so here is a syllabus excerpt. This is for a Calculus class, using Folland's Text, material covered was as follows on the test:

  1. Differential Calculus. Differentiability in One Variable. Differentiability in Several Variables. The Chain Rule. The Mean Value Theorem. Functional Relations and Implicit Functions: A First Look. Higher-Order Partial Derivatives. Taylor's Theorem. Critical Points. Extreme Value Problems. Vector-Valued Functions and Their Derivatives.

  2. The Implicit Function Theorem and Its Applications. The Implicit Function Theorem. Curves in the Plane. Surfaces and Curves in Space. Transformations and Coordinate Systems. Functional Dependence.

  3. Integral Calculus. Integration on the Line. Integration in Higher Dimensions. Multiple Integrals and Iterated Integrals. Change of Variables for Multiple Integrals. Functions Defined by Integrals. Improper Integrals. Improper Multiple Integrals. Lebesgue Measure and the Lebesgue Integral.

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An arbitrary point of the surface is $(x,y,xy)$. The distance, $D$ from $(1,1,1)$ to this point is $D=\sqrt{(1-x)^2+(1-y)^2+(1-xy)^2}$. You need to find the minimum value of $D$ (using gradient methods, e.g.). By the way, that point is on the surface; so the shortest distance is 0... –  David Mitra Nov 15 '11 at 9:26

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