Let f(x, y) = x 2 + y 2 . (a) Using the method of Lagrange multipliers, optimize

Question

Let f(x, y) = x 2 + y 2 .

(a) Using the method of Lagrange multipliers, optimize f(x, y) with respect to the constraint x + y = 10. That is, find the point (x, y) satisfying the constraint at which f(x, y) has an extremum.

(b) Is the extremum you found in part (a) a maximum or minimum of f(x, y) relative to the constraint? Explain. It may be fruitful to try part (c) first or to solve part (a) again by using the constraint to replace one of the variables as in 155.

(c) Explain part (a) graphically. Hint: think about the graph of f(x, y) and what its intersection with x + y = 10 looks like in R 3

Explanation / Answer

by lagrange multipliers the extremum is (5,5)

the extremum is at (5,5) is minumum

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