Math 142 Lecture Notes for Section 8.3
Section 8.3 -
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Extrema of Functions of Two Variables
Definition 8.3.1:
Let z = f (x, y) be a function of two variables. The value f (a, b) is
(a) a local maximum if there exists a circular region in the domain of f with (a, b) as
the center such that f (a, b) ≥ f (x, y) for all (x, y) in the region.
(b) a local minimum if there exists a circular region in the domain of f with (a, b) as
the center such that f (a, b) ≥ f (x, y) for all (x, y) in the region.
Definition 8.3.2:
A critical point of z = f (x, y) is a where both first partial derivatives are zero:
Example 8.3.3:
Determine the critical points of 4x3 + 3y + 2xy + 4y 2 − 8x + 3
Math 142 Lecture Notes for Section 8.3
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Definition 8.3.4:
∂ 2f ∂ 2f
∂ 2f
Let z = f (x, y) be a function of two variables such that
exist for
,
, and
∂x2 ∂y 2
∂xy
∂f
= 0 and
every point inside a circle centered at (a, b). If (a, b) is a critical point
∂x
∂f
= 0, we define a number D to be
∂y
2 2
∂ 2f ∂ 2f
∂ f
D=
· 2 −
.
2
∂x ∂y
∂xy
Then,
(1) If D(a, b) > 0 and
∂ 2f
< 0, then f has a
∂x2
at (a, b).
(2) If D(a, b) > 0 and
∂ 2f
> 0, then f has a
∂x2
at (a, b).
(3) If D(a, b) < 0 then f has a
at (a, b).
(4) If D(a, b) = 0 then no conclusion can be made about f (a, b).
Example 8.3.5:
Find all critical points and determine whether each is a saddle point, local max, or local
min.
(a) f (x, y) = −x2 − y 2 + 6x + 8y − 21
Math 142 Lecture Notes for Section 8.3
(b) f (x, y) = x3 + y 3 − 6xy
Suggested Homework Problems: 1-15(odd), 19, 25, 27, 33
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