Example 1 Find
the absolute minimum and absolute maximum of on the rectangle given by and .
Solution
Let’s first get a quick picture of the rectangle for
reference purposes.
The boundary of this rectangle is given by the following
conditions.
These will be important in the second step of our process.
We’ll start this off by finding all the critical points
that lie inside the given rectangle.
To do this we’ll need the two first order derivatives.
Note that since we aren’t going to be classifying the
critical points we don’t need the second order derivatives. To find the critical points we will need to
solve the system,
We can solve the second equation for y to get,
Plugging this into the first equation gives us,
This tells us that we must have or . Now, recall that we only want critical
points in the region that we’re given.
That means that we only want critical points for which . The only value of x that will satisfy this is the first one so we can ignore the
last two for this problem. Note
however that a simple change to the boundary would include these two so don’t
forget to always check if the critical points are in the region (or on the
boundary since that can also happen).
Plugging into the equation for y gives us,
The single critical point, in the region (and again,
that’s important), is . We now need to get the value of the
function at the critical point.
Eventually we will compare this to values of the function
found in the next step and take the largest and smallest as the absolute
extrema of the function in the rectangle.
Now we have reached the long part of this problem. We need to find the absolute extrema of the
function along the boundary of the rectangle.
What this means is that we’re going to need to look at what the
function is doing along each of the sides of the rectangle listed above.
Let’s first take a look at the right side. As noted above the right side is defined by
Notice that along the right side we know that . Let’s take advantage of this by defining a
new function as follows,
Now, finding the absolute extrema of along the right side will be equivalent to
finding the absolute extrema of in the range . Hopefully you recall how to do this from Calculus
I. We find the critical points of in the range and then evaluate at the critical points and the end points of
the range of y’s.
Let’s do that for this problem.
This is in the
range and so we will need the following function evaluations.
Notice that, using the definition of these are also function values for .
We can now do the left side of the rectangle which is
defined by,
Again, we’ll define a new function as follows,
Notice however that, for this boundary, this is the same
function as we looked at for the right side.
This will not always happen, but since it has let’s take advantage of
the fact that we’ve already done the work for this function. We know that the critical point is and we know that the function value at the
critical point and the end points are,
The only real difference here is that these will
correspond to values of at different points than for the right
side. In this case these will
correspond to the following function values for .
We can now look at the upper side defined by,
We’ll again define a new function except this time it will
be a function of x.
We need to find the absolute extrema of on the range . First find the critical point(s).
The value of this function at the critical point and the
end points is,
and these in turn correspond to the following function
values for
Note that there are several “repeats” here. The first two function values have already
been computed when we looked at the right and left side. This will often happen.
Finally, we need to take care of the lower side. This side is defined by,
The new function we’ll define in this case is,
The critical point for this function is,
The function values at the critical point and the endpoint
are,
and the corresponding values for are,
The final step to this (long…) process is to collect up
all the function values for that we’ve computed in this problem. Here they are,
The absolute minimum is at since gives the smallest function value and
the absolute maximum occurs at and since these two points give the largest
function value.
Here is a sketch of the function on the rectangle for
reference purposes.
