Maximum and Minimum Values
Continuous functions of two variables assume extreme values on
closed, bounded domains. We see in this lecture that we can
narrow the search for these extreme values by examining the
functions’ first partial derivatives.
Derivative Tests for Local Extreme Values
Definition
Let f (x, y ) be defined on a region R containing the point (a, b).
Then
1
f (a, b) is a local maximum value of f if f (a, b) ≥ f (x, y ) for
all points (x, y ) in an open disk centered at (a, b).
2
f (a, b) is a local minimum value of f if f (a, b) ≤ f (x, y ) for
all points (x, y ) in an open disk centered at (a, b).
Derivative Tests for Local Extreme Values
Theorem
If f (x, y ) has a local maximum or minimum value at a point (a, b)
and if the first partial derivatives exists there, then fx (a, b) = 0 and
fy (a, b) = 0.
This theorem says that the surface has a horizontal tangent plane
at a local extremum, provided there is a tangent plane there.
Derivative Tests for Local Extreme Values
Definition
A point in the domain of a function f (x, y ) where both fx and fy or
where one or both of fx and fy do not exist is a critical point of f .
As with differentiable functions of one variable, not every critical
point gives rise to a local extremum.
Derivative Tests for Local Extreme Values
Definition
A differentiable function f (x, y ) has a saddle point at a critical
point (a, b) if in every open disk centered at (a, b) there are
domain points (x, y ) where f (x, y ) > f (a, b) and domain points
(x, y ) where f (x, y ) < f (a, b).
Derivative Tests for Local Extreme Values
Example
Find the local extreme values (if any) of f (x, y ) = y 2 − x 2 .
Derivative Tests for Local Extreme Values
Theorem (Second Derivative Test for Local Extreme Values)
Suppose that f (x, y ) and its first and second partial derivatives are
continuous throughout a disk centered at (a, b) and that
fx (a, b) = fy (a, b) = 0. Then
1
f has a local maximum at (a, b) if fxx < 0 and
fxx fyy − fxy2 > 0 at (a, b).
2
f has a local minimum at (a, b) if fxx > 0 and
fxx fyy − fxy2 > 0 at (a, b).
3
f has a saddle point at (a, b) if fxx fyy − fxy2 < 0 at (a, b).
4
the test is inconclusive at (a, b) if fxx fyy − fxy2 = 0 at (a, b).
In this case, we must find some other way to determine the
behaviour of f at (a, b).
Derivative Tests for Local Extreme Values
The expression fxx fyy − fxy2 is called the discriminant or Hessian
of f . It is sometimes easier to remember it in determinant form.
fxx fyy − fxy2 =
fxx
fxy
fxy
fyy
Derivative Tests for Local Extreme Values
Example
Find the local extreme values of the function
f (x, y ) = xy − x 2 − y 2 − 2x − 2y + 4.
Derivative Tests for Local Extreme Values
Example
Find the local extreme values of
f (x, y ) = 3y 2 − 2y 3 − 3x 2 + 6xy .
Absolute Maxima and Minima on Closed Bounded Regions
A closed set in R2 is one that contains all its boundary points.
A bounded set in R2 is one that is contained within some disk.
Absolute Maxima and Minima on Closed Bounded Regions
We organize the search for absolute extrema of a continuous
function f (x, y ) on a closed and bounded set D into three steps.
1
Find the values of f at the critical points of f in D.
2
Find the extreme values of f on the boundary of D.
3
The largest value from steps 1 and 2 is the absolute maximum
value; the smallest of these is the absolute minimum value.
Absolute Maxima and Minima on Closed Bounded Regions
Example
Find the absolute maximum and minimum values of
f (x, y ) = 2 + 2x + 2y − x 2 − y 2
on a triangular region in the first quadrant bounded by the lines
x = 0, y = 0, y = 9 − x.