Multivariate Calculus - Bruce E. Shapiro

LECTURE 14. UNCONSTRAINED OPTIMIZATION 105
The Second Derivative Test: Classifying the Stationary
Points
Theorem 14.2 Second Derivative Test.Suppose that f(x, y) has continuous partial
derivatives in some neighborhood of (a, b) where
Define the function
and the number
Then
f x (a, b) = f y (a, b) = 0
D(x, y) = f xx f yy − f 2 xy
d = D(a, b) = f xx (a, b)f yy (a, b) − (f xy (a, b)) 2
1. If d > 0 and f xx (a, b) < 0, then f(a, b) is a local maximum;
2. If d > 0 and f xx (a, b) > 0, then f(a, b) is a local minimum;
3. If d < 0 then f(a, b) is a saddle point;
4. If d = 0 the second-derivative test is inconclusive.
Proof of the Second Derivative Test. Given at the end of this section.
Example 14.3 Classify the stationary points of f(x, y) = x 3 + y 3 − 6xy
Solution. We found in the previous example that stationary points occur at (0, 0)
and (2, 2), and that
f x = 3x 2 − 6y
Hence the second derivatives are
Thus
At (a, b) = (0, 0),
f y = 3y 2 − 6x
f xx = 6x
f xy = −6
f yy = 6y
D(x, y) = f xx f yy − f 2 xy = (6x)(6y) − (−6) 2 = 36xy − 36
hence (0, 0) is a saddle point.
and
At (a, b) = (2, 2),
Hence (2, 2) is a local minimum.
d = −36 < 0
d = (36)(2)(2) − 36 > 0
f xx 2, 2 = 12 > 0
Math 250, Fall 2006 Revised December 6, 2006.