(3pt) For a small displacement (∆x, ∆y) the change in the function f (x, y)
at a fixed point (∇f = (fx, fy ) = 0) can be approximated by:
∆f = ∆x ∆y fxx fxy
fxy fyy
∆x
∆y
Show,
(a) ∆f > 0 for any non-zero displacement (∆x, ∆y) when fxxfyy - f 2
xy > 0
and fxx > 0.
(b) ∆f < 0 for any non-zero displacement (∆x, ∆y) when fxxfyy - f 2
xy > 0
and fxx < 0.
(c) The sign of ∆f depends on the displacement (∆x, ∆y) when fxxfyy -
f 2
xy < 0.
Hint: Recall that the solution to the quadratic equation: au2 + bu + c = 0, is
u = (-b\pm \sqrt(b)2 - 4ac)/(2)a. In particular a solution doesn’t exist if b2 -4ac < 0,
which means that the parabola y = au2 + bu + c never crosses the u axis.