Find the linearization Lx y of fx y x2 xy 12 y2 3 at the

Find the linearization L(x, y) of f(x, y) = x^2 - xy + 1/2 y^2 + 3 at the point (3,2).

The Linearization of f(x,y) about the point (a,b) is L(x,y) = f(a,b) + fx(a,b) (xa) + fy(a,b) (yb).-------(2)

The partial derivatives of f(x,y) w.r.t x and y are fx(x,y) =2x-y and fy(x,y)=-x+y

Substituting x=3 and y =2 in f(x,y),fx(x,y) and fy(x,y), then we get

f(3,2)=8, fx(3,2)=4,fy(3,2) =1

The Linearization of f(x,y) about the point (3,2) is L(x,y) = f(3,2) + fx(3,2) (x3) + fy(3,2) (y2).

Hence, L(x,y) = 8 + 4 (x3) 1 (y2).