Math 275 Notes
Topic 3.7: Critical Points, Local Extrema, and the
Second Derivative Test for Functions of Two
Variables
Textbook Section: 14.7
From the Toolbox (what you need from previous classes):
Algebra: Solving systems of two equations in two variables.
Calc I: Critical points, local extrema (maxima and minima) and the second
derivative test for functions of a single variable (from Calc I).
Calc III: Computing and evaluating first and second partial derivatives
and gradients of functions of two variables. Familiarity with level curves
and contour maps.
Learning Objectives (New Skills) & Important Concepts
Learning Objectives (New Skills):
Find critical points of functions of two variables.
Use contour maps to determine whether a point is a local maximum,
local minimum, or saddle point.
Use the second derivative test to determine whether critical points
are local minima, local maxima, or saddle points.
Important Concepts:
Critical points occur when ∇f (x, y ) = ~0 (a horizontal tangent
plane), or when one or both of the partial derivatives does not exist
(no tangent plane).
Local maxima and minima occur only at critical points.
Not every critical point corresponds to a local maximum or minimum.
For example: saddle points can occur at critical point. (Saddle
points are two-dimensional versions of inflection points.)
For a critical point (a, b) where ∇f (a, b) = ~0 and D(a, b) 6= 0, the
second derivative test will determine whether there is a local maximum, local minimum, or saddle point at (a, b).
The Big Picture
The theory and application of critical points and local extrema for functions
of two variables are directly analogous to those for functions of a single variable.
For functions of a single variable, critical points occur when the derivative equals
zero, or does not exist. For functions of two variables, critical points occur when
either both partial derivatives equal zero (which is the same as saying ∇f = ~0)
or when one or both of the partial derivatives do not exist.
For both single-variable and bivariate functions, local maximum and minimum
values (extrema) can only occur at critical points. Also in both cases, a “second derivative test” can be used to determine whether critical points are local
maxima, local minima, or inflection/saddle points.
More Details
◦ Definitions of local minimum, local maximum, and saddle points for a
function f (x, y ):
f (a, b) is a local maximum if:
f (a, b) ≥ f (x, y )
for all points (x, y ) near the point (a, b).
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f (a, b) is a local minimum if:
f (a, b) ≤ f (x, y )
for all points (x, y ) near the point (a, b).
(a, b) is a saddle point if:
◦ ∇f (a, b) = ~0
◦ f (a, b) is neither a local maximum nor a
local minimum.
(No matter how close you get to (a, b),
there are some points (x, y ) such that
f (a, b) > f (x, y ), and others such that
f (a, b) < f (x, y ).)
◦ The second derivative test for functions of two variables may look more
complicated than the second derivative test for a function of a single
variable, but it really measures the same thing:
? The Calc I second derivative test tells you whether the graph of the
function y = f (x) lies entirely on one side of the tangent line.
? The Calc III second derivative test tells you whether the graph of
the function z = f (x, z) lies entirely on one side of the tangent
plane.
◦ There are two instances in which the second derivative test does not
work, and you need to try something else to determine whether there is
a local max or min at a critical point (a, b). These two instances are:
? The critical point occurs because one or both of the partial derivatives does not exist. In this case, the function D can’t be used,
since you need partial derivatives for D.
? If D(a, b) = 0, the test fails.
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Recall from Calc I: These are the same two conditions that will cause the
second derivative test to fail.
Using the Second Derivative Test
The second derivative test can be used when both of these conditions
are met:
i. ∇f (a, b) = ~0.
ii. The second partial derivatives of f (x, y ) exist and are continuous
near (a, b).
The function used to determine the nature of the critical point (a, b) is:
h
i2
D(a, b) = fxx (a, b)fy y (a, b) − fxy (a, b)
or, in Leibnitz notation:
2 2
∂ f
∂ 2f ∂ 2f
−
D(a, b) =
2
2
∂x ∂y
∂y ∂x
The test works as follows:
? If D(a, b) > 0 and fxx (a, b) < 0, then f (a, b) is a local maximum.
? If D(a, b) > 0 and fxx (a, b) > 0, then f (a, b) is a local minimum.
? If D(a, b) < 0, then f has a saddle point at (a, b).
Note: If D(a, b) = 0, the test is inconclusive. You need to find another
way to determine whether f (a, b) is a local maximum, minimum, or saddle point.
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