Calculus 2nd Edition Rogawski

SECTION 15.7 Optimization in Several Variables 841
To solve these equations, set the numerators equal to zero. Figure 4 suggests that f (x, y)
has a local max with x>0 and a local min with x 1.095, y -> -0.274}
Thus, (1.095, −0.274) is an approximate critical point where, by Figure 4, f takes on a
local maximum.Asecond search near (−1, 0) yields (−1.095, 0.274), which approximates
the critical point where f (x, y) takes on a local minimum.
We know that in one variable, a function f (x) may have a point of inflection rather
than a local extremum at a critical point.Asimilar phenomenon occurs in several variables.
Each of the functions in Figure 5 has a critical point at (0, 0). However, the function in
Figure 5(C) has a saddle point, which is neither a local minimum nor a local maximum.
If you stand at the saddle point and begin walking, some directions take you uphill and
other directions take you downhill.
z
z
z
y
y
y
x
x
x
FIGURE 5
(A) Local maximum (B) Local minimum (C) Saddle
The discriminant is also referred to as the
“Hessian determinant.”
As in the one-variable case, there is a Second Derivative Test for determining the type
of a critical point (a, b) of a function f (x, y) in two variables. This test relies on the sign
of the discriminant D = D(a,b), defined as follows:
D = D(a,b) = f xx (a, b)f yy (a, b) − fxy 2 (a, b)
If D>0, then f xx (a, b) and f yy (a, b)
must have the same sign, so the sign of
f yy (a, b) also determines whether f (a, b)
is a local minimum or a local maximum.
THEOREM 2 Second Derivative Test Let P = (a, b) be a critical point of f (x, y).
Assume that f xx ,f yy ,f xy are continuous near P . Then:
(i) If D>0 and f xx (a, b) > 0, then f (a, b) is a local minimum.
(ii) If D>0 and f xx (a, b) < 0, then f (a, b) is a local maximum.
(iii) If D