# Help! exam tomorrow and stuck on this proof

1. May 25, 2009

### butters120

i have a math method exam tomorrow, and im really stuck on this proof, im not sure how they have manipulated the 2nd bracket to get the other version of this equation. any help would be much appreciated.

To establish just what constitutes sufficient conditions we first note that, since
f is a function of two variables and ∂f/∂x = ∂f/∂y = 0, a Taylor expansion of
the type (5.18) about the stationary point yields
f(x, y) − f(x0, y0) ≈1/2! [(Δx)2fxx + 2ΔxΔyfxy + (Δy)2fyy]

where Δx = x − x0 and Δy = y − y0 and where the partial derivatives have been
written in more compact notation. Rearranging the contents of the bracket as
the weighted sum of two squares, we find
f(x, y) − f(x0, y0) ≈1/2![fxx(Δx + fxyΔy/fxx)^2 +(Δy)^2(fyy −f^2xy/fxx]

thanks