Critical
Points of a Real-Valued Function of Two Variables by Tom Davis1
At a
critical
or stationary point on a surface, the tangent plane is horizontal, analogous to a horizontal
tangent line for critical points on a curve.
Given a twice continuously differentiable or
C2,
real-valued function z of a vector variable (x,y), denote the partial derivative
of z with respect to x as zx and the partial
derivative of z with respect to y as zy. Then the
gradient of z is the vector grad
z = (zx,zy).
Analogous to to the necessary condition
that the first derivative vanish for critical points on a curve, a point
(x0,y0)
is a critical point if, and only if, grad z(x0,y0)
= (0,0).
Denote the partial derivatives of zx
and zy as zxx,
zxy = zyx,
and zyy. Then the
Hessian matrix2
of z is the 2 x 2 symmetric matrix H(z) = [hi j(x,y)]
where h11
= zxx, h12
= zxy = h21,
and h22 = zyy. The
Hessian determinant of z is the
discriminant
D = |H| = h11 h22
- h122.
Analogous to the second derivative test
for critical points on a curve, a critical point (x0,y0)
is a
