Determining the maximum and minimum values of function

Question

Determine the maximum and minimum values of the function f(x,y)=x^2+y^2-2x-4y+5 in the set D = {( x ,y )∈R^2 / x^2 + y^2 ≤ 45}.

Answer 1

x^2 -2x +y^2 -4y +5=0 complete the squares

x^2 -2x +1 +y^2 -4x +4 =-5 +1+4 =0

(x-1)^2 + (y-2)^2 = 0 is a dot, one point (1,2) a degenerate "circle" with radius =0

and center = (1,2)= maximum = minimum. use a graphing calculator

if nothing seems to show up, graph x^2+y^2-2x-4y+4.999=0 using 4.999 in place of 5

Answer 2

Find local max and mins inside D using first and second partial derivatives.

Find max and mins on the boundary of D (a circle of radius root(45)) using Lagrange multipliers.

The absolute max is the largest of all found points and the absolute min is the smallest of all found points.