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In this section we are going to extend one of the more
important ideas from Calculus I into functions of two variables. We are going to start looking at trying to
find minimums and maximums of functions.
This in fact will be the topic of the following two sections as well.
In this section we are going to be looking at identifying
relative minimums and relative maximums.
Recall as well that we will often use the word extrema to refer to both
minimums and maximums.
The definition of relative extrema for functions of two
variables is identical to that for functions of one variable we just need to
remember now that we are working with functions of two variables. So, for the sake of completeness here is the
definition of relative minimums and relative maximums for functions of two
variables.
Definition
Note that this definition does not say that a relative
minimum is the smallest value that the function will ever take. It only says that in some region around the
point 
the function will always be larger than . Outside of that region it is completely
possible for the function to be smaller.
Likewise, a relative maximum only says that around the function will always be smaller than . Again, outside of the region it is completely
possible that the function will be larger.
Next we need to extend the idea of critical points up to functions of two variables. Recall that a critical point of the function was a number so that either or doesn’t exist.
We have a similar definition for critical points of functions of two
variables.
Definition
To see the equivalence in the first part let’s start off
with and put in the definition of each part.
The only way that these two vectors can be equal is to
have and ). In fact, we will use this definition
of the critical point more than the gradient definition since it will be easier
to find the critical points if we start with the partial derivative definition.
Note as well that BOTH of the first order partial
derivatives must be zero at . If only one of the first order partial
derivatives are zero at the point then the point will NOT be a critical point.
We now have the following fact that, at least partially,
relates critical points to relative extrema.
Fact
Proof
Note that this does NOT say that all critical points are
relative extrema. It only says that
relative extrema will be critical points of the function. To see this let’s consider the function
The two first order partial derivatives are,
The only point that will make both of these derivatives zero
at the same time is and so is a critical point for the function. Here is a graph of the function.
Note that the axes are not in the standard orientation here
so that we can see more clearly what is happening at the origin, i.e. at . If we start at the origin and move into
either of the quadrants where both x
and y are the same sign the function
increases. However, if we start at the
origin and move into either of the quadrants where x and y have the opposite
sign then the function decreases. In
other words, no matter what region you take about the origin there will be
points larger than and points smaller than