identify any extrema of the function by recognizing its given form or its form after completing the square…

Explanation:

Step1: Analyze the function form

The function (g(x,y)=(x - 2)^2+(y - 3)^2) is in the form of the sum of two squares. Since ((x - 2)^2\geq0) and ((y - 3)^2\geq0) for all real (x) and (y), the minimum value of (g(x,y)) occurs when (x - 2=0) and (y - 3=0). When (x = 2) and (y=3), (g(2,3)=(2 - 2)^2+(3 - 3)^2=0). For any other values of (x) and (y), ((x - 2)^2>0) or ((y - 3)^2>0), so (g(x,y)>0). Since the function is a sum of non - negative terms, it has no maximum (as (x\to\pm\infty) or (y\to\pm\infty), (g(x,y)\to+\infty)).

Step2: Use partial derivatives (verification)

First, find the first - order partial derivatives: The partial derivative with respect to (x) is (g_x(x,y)=2(x - 2)) Set (g_x(x,y)=0), then (2(x - 2)=0), which gives (x = 2) The partial derivative with respect to (y) is (g_y(x,y)=2(y - 3)) Set (g_y(x,y)=0), then (2(y - 3)=0), which gives (y = 3) The second - order partial derivatives: (g_{xx}(x,y)=2), (g_{yy}(x,y)=2), (g_{xy}(x,y)=0) The discriminant (D=g_{xx}(x,y)g_{yy}(x,y)-[g_{xy}(x,y)]^2=(2)(2)-0^2 = 4>0) and (g_{xx}(2,3)=2>0)

Answer:

relative minimum ((x,y,z)=(2,3,0)); relative maximum: DNE