Multivariate Calculus - Bruce E. Shapiro

102 LECTURE 14. UNCONSTRAINED OPTIMIZATION
1. All points on the boundary of J;
2. All stationary points, namely, any point where ∇f(x, y) = 0;
3. All singular points, namely, any points where f(x, y) or ∇f(x, y) are undefined.
The candidate extrema are sometimes called critical points; some authors restrict
the use of this term to describe the stationary points.
Stationary points occur when ∇f(x, y) = 0; this requires all of the derivatives
to be zero:
∂f
∂x = 0 = ∂f
∂y
Recall from one-variable calculus that three different things could happen at a stationary
point where f ′ (a) = 0:
• f(x) could have a local maximum at x = a, as, for example, occurs for f(x) =
−(x − a) 2
• f(x) could have a local minimum at x = a, as, for example, occurs for f(x) =
(x − a) 2
• f(x) could have an inflection point at x = a, as, for example, occurs for
f(x) = (x − a) 3
For multivariate functions there are three types of stationary points: local maxima,
local minima, and saddle points. Saddle points are the generalization of inflection
points. At a saddle point, in some vertical cross-sections, the function appears to
have a local minimum, while in other vertical cross-sections, the function appears
to have a local maximum.
Example 14.1 Find the stationary points of the function f(x, y) = x 2 − 7xy +
12y 2 − y
Solution. The function has stationary points when f x (x, y) = f y (x, y) = 0. Differentiating,
∂f
∂x = 0 ⇒ ∂
∂x (x2 − 7xy + 12y 2 − y) = 2x − 7y = 0
⇒ y = 2x/7
∂f
∂y = 0 ⇒ ∂ ∂y (x2 − 7xy + 12y 2 − y) = −7x + 24y − 1 = 0
⇒ 24y = 7x + 1
Combining the two results,
48
49x + 7
x = 7x + 1 = ⇒ x = −7
7 7
Revised December 6, 2006. Math 250, Fall 2006