The Extreme Value Theorem for functions of two variables states that if a real-valued function f(x,y)f of open paren x comma y close paren 𝑓(𝑥,𝑦)
is continuous on a closed and bounded set Dcap D
𝐷
in R2R-2
ℝ2
, then ff
𝑓
must attain an absolute maximum and an absolute minimum value on Dcap D
𝐷
. The absolute maximum and minimum values are the highest and lowest points on the surface defined by the function over the specified region.
Core concepts and conditions
The Extreme Value Theorem for functions of two variables relies on two critical conditions:
-
Continuity of the function: The function f(x,y)f of open paren x comma y close paren
𝑓(𝑥,𝑦)
must be continuous on the set Dcap D
𝐷
. This means there are no breaks, jumps, or holes in the surface defined by the function over the region.
-
**A closed and bounded set Dcap D
𝐷** : The domain Dcap D
𝐷
must be both closed and bounded.
-
Closed set: A set is closed if it contains its entire boundary. In R2R-2
ℝ2
, this means the region includes all the points on its edge.
-
Bounded set: A set is bounded if it is contained within some disk of finite radius. Intuitively, this means the region does not extend to infinity in any direction.
-
Intuitive explanation
Imagine a continuous surface, like a mountain range, over a specific area on a map. If the area on the map is a square (closed and bounded), the Extreme Value Theorem guarantees that there will be a highest peak and a lowest valley within that square, including its edges. If the region were not bounded (e.g., the entire infinite plane), or not closed (e.g., the interior of the square without its edges), this guarantee would no longer hold. The mountain range might get infinitely tall or drop infinitely low, or the peak might be on an edge that isn't included in the domain.
Finding absolute extrema using the Extreme Value Theorem
The theorem guarantees the existence of absolute extrema, but it does not specify their location. To find them, you must test all possible candidates. The absolute maximum and minimum values of f(x,y)f of open paren x comma y close paren
𝑓(𝑥,𝑦)
on a closed and bounded set Dcap D
𝐷
can occur at one of two places:
-
Critical points in the interior of the set Dcap D
𝐷
.
-
Boundary points of the set Dcap D
𝐷
.
This gives a straightforward, multi-step process for finding the absolute extrema:
Step 1: Find critical points
-
Calculate the first-order partial derivatives of f(x,y)f of open paren x comma y close paren
𝑓(𝑥,𝑦)
, denoted as fxf sub x
𝑓𝑥
and fyf sub y
𝑓𝑦
.
-
Solve the system of equations fx(x,y)=0f sub x of open paren x comma y close paren equals 0
𝑓𝑥(𝑥,𝑦)=0
and fy(x,y)=0f sub y of open paren x comma y close paren equals 0
𝑓𝑦(𝑥,𝑦)=0
simultaneously.
-
Identify any points (x,y)open paren x comma y close paren
(𝑥,𝑦)
that satisfy the equations and lie within the interior of the domain Dcap D
𝐷
. These are your critical point candidates.
Step 2: Investigate the boundary
-
This is often the most complex part of the process, as it involves reducing a 2D problem to a 1D problem.
-
The boundary of the set Dcap D
𝐷
is a curve (or set of curves). You must find the extreme values of f(x,y)f of open paren x comma y close paren
𝑓(𝑥,𝑦)
along this boundary.
-
This can be done by parameterizing the boundary curve(s) and using single-variable calculus to find critical points and endpoints for each segment.
-
Evaluate the original function ff
𝑓
at all of these boundary candidates.
Step 3: Compare all candidates
-
After finding all candidate points from Step 1 (interior critical points) and Step 2 (boundary candidates), evaluate the original function ff
𝑓
at each point.
-
The largest value is the absolute maximum, and the smallest value is the absolute minimum of the function on the set Dcap D
𝐷
.
A note on the generalized theorem
The Extreme Value Theorem is a powerful result from real analysis that extends beyond functions of two variables. It states that any continuous, real-valued function on a non-empty compact space (a space that is closed and bounded) will attain a maximum and a minimum value. In RnR-n
ℝ𝑛
, the Heine-Borel theorem proves that a set is compact if and only if it is closed and bounded. Therefore, the Extreme Value Theorem for functions of two variables (or any number of variables) is a direct consequence of this more general theorem.