Calculus: Find the Local Maxima, Minima, and Saddle Points

Find the local maxima, minima, and saddle points for f(x,y) = x^2 – 2xy + 2y^2 – 2x + 2y + 1

Start by taking the partial derivatives fx and fy

fx = 2x – 2y – 2

fy = -2x + 4y + 2

Because both functions exist at all vallues (x,y), a local extreme can only occur where

fx = 2x – 2y – 2 = 0 and fy = -2x + 4y + 2 = 0

Possibilities: (1,0)

At (1,0):

fxx = 2, fyy = 4, fxy^2 = 4, and fxxfyy – fxy^2 = 4

Solution: By the Second Derivative Test for Local Extremes, we find that there exists:

Local minima at (1,0) with a value of 0