8. Functions of Several Variables
8.3: Locating Extrema of Functions of Two Variables
Let f (x, y) be a function of two variables. The value f (a, b) is
• a relative (local) maximum of f at (a, b) if f (a, b) ≥ f (x, y) for all points (x, y) in some disk
centered at (a, b).
• a relative minimum of f at (a, b) if f (a, b) ≤ f (x, y) for all points (x, y) in some disk centered
at (a, b).
• an absolute maximum of f at (a, b) if f (a, b) ≥ f (x, y) for all points (x, y) in the domain of f .
• an absolute minimum of f at (a, b) if f (a, b) ≤ f (x, y) for all points (x, y) in the domain of f .
• a relative extremum of f at (a, b) if it is either a relative maximum or a relative minimum at
(a, b).
• an absolute extremum of f at (a, b) if it is either an absolute maximum or an absolute minimum
at (a, b).
Necessary Condition for Relative Extrema: If z = f (x, y) is defined, both first-order partial
derivatives exist for all (x, y) in some disk centered at (a, b), and f has a relative extremum at (a, b),
then
fx (a, b) = 0, and fy (a, b) = 0.
Geometrically, if f has a tangent plane at (a, b) and f has a local extremum at (a, b), then the tangent
plane must be horizontal.
Critical Point
Assume that both first-order partial derivatives of z = f (x, y) exist in the domain of f . A critical
point (a, b) of z = f (x, y) is a point not on the boundary of the domain of f (x, y), for which the two
partial derivatives vanish:
fx (a, b) = 0, and fy (a, b) = 0.
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Remark: If f has a local extremum at (a, b), then (a, b) is a critical point of f . However, not all
critical points give rise to extrema. At a critical point, a function may have a local maximum or a local
minimum or neither.
A critical point that is neither a local maximum nor a local minimum is called a saddle point.
Example 1. Find the critical points of each of the following functions.
a) x4 + y 4 − x2 − 2xy − y 2
b) x3 + 3xy 2 − 15x − 12y
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c) g(s, t) = s2 + t2 + 2t − 4s + 1
d) h(x, y) = 2x2 − y 2 + 4xy + 2x − 2y
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e) f (x, y) = x3 + 6xy + y 3
f ) f (x, y) =
x2 y 2 − 27x + y
xy
g) g(u, v) = u2 + v 2 + u2 v + v 2 u
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Classify the critical points (if any) of a function
Second Derivative Test for Functions of Two Variables:
Given z = f (x, y) a function with fxx , fyy , and fxy continuous at every point inside a disk centered at
(a, b). If (a, b) is a critical point, i.e.
fx (a, b) = 0 and fy (a, b) = 0,
we define the number D(a, b) by
D(a, b) = fxx (a, b)fyy (a, b) − [fxy (a, b)]2 .
Then,
1. If D(a, b) > 0 and fxx < 0, f has a relative maximum at (a, b).
2. If D(a, b) > 0 and fxx > 0, f has a relative minimum at (a, b).
3. If D(a, b) < 0, (a, b) is a saddle point (at which f has neither a relative maximum nor relative
minimum).
4. If D(a, b) = 0, no conclusion about f (a, b).
Example 2. For each function in example 1, find all critical points and classify.
a) x4 + y 4 − x2 − 2xy − y 2
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b) x3 + 3xy 2 − 15x − 12y
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c) g(s, t) = s2 + t2 + 2t − 4s + 1
d) h(x, y) = 2x2 − y 2 + 4xy + 2x − 2y
e) f (x, y) = x3 − 3xy + y 3
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f ) f (x, y) =
x2 y 2 − 27x + y
xy
g) g(u, v) = u2 + v 2 + u2 v + v 2 u
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Example 3. (Brief Calculus, Armstrong and Davis) New Jeans, a specialty blue jeans manufacturer,
produces two types of blue jeans each day, x pairs of straight-leg jeans and y pairs of wide-leg jeans.
The daily profit function, in dollars, is given by
P (x, y) = 78x + 3xy − 3x2 − y 2 − 2y.
a) How many straight-leg jeans and how many wide-leg jeans should be produced and sold each day to
maximize profit?
b) What is the maximum profit?
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