The function f(x, y) = x^2 + y^2 - 4x - 6y + 13 has a local extremum at the point (a, b). What is the point and the nature of the extremum?

(A) (2, 3) is a local minimum with value 0

(B) (2, 3) is a local maximum with value 13

(C) (-2, -3) is a saddle point

(D) (4, 6) is a local minimum

Answer Key:

A

### Solution: f_x = 2x - 4 = 0 ==> x = 2. f_y = 2y - 6 = 0 ==> y = 3. f_{xx} = 2 > 0, f_{yy} = 2 > 0, f_{xy} = 0. Discriminant D = 2(2) - 0 = 4 > 0 and f_{xx} > 0 ==> Local minimum. Value: 2^2 + 3^2 - 4(2) - 6(3) + 13 = 4 + 9 - 8 - 18 + 13 = 0. Correct: A.

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