We need to determine the absolute extrema on the closed interval from negative one to
four of the given function. Now, absolute extrema are different from relative extrema,
so let's start with a quick review. To determine the absolute extrema on a closed interval,
we want to determine the largest function value and the smallest function value for
all values of X in the interval. One of the biggest differences between absolute
extrema and relative extrema is that absolute extrema can occur at the end points of the
closed interval. So to determine the largest and smallest function values, we'll have to
evaluate the function at the critical numbers as well as the end points of the interval.
Looking at this function on this closed interval, notice that the largest function value occurs at
this point here, at the end point, so this point here represents the absolute maximum
on this closed interval. And notice how the least function value is negative one at X
equals negative two, so this point here represents the absolute minimum. Again, notice that it occurs
at the end points. Now notice how these two points do represent relative extrema,
but they are not absolute extrema because there are function values that are greater than this
function value here, and there are function values that are less than this function value here.
Now looking at the second graph, notice that this relative minimum here is also
the absolute minimum because it is the least function value on this closed interval. But
the absolute maximum would occur at X equals six with a function value of positive five.
So the idea here is to determine the absolute maximum and minimum, we do have to determine
the function values of the critical numbers and the end points. So let's give it a try.
So we'll start by determining the critical numbers of the given function. So we'll
determine the first derivative. Here we would have X cubed minus three X squared. Now we'll
set this equal to zero and solve. The greatest common factor here would be X squared, leaving
us with the quantity X minus three. So our critical numbers are X equals zero and
X equals three. So now to determine the absolute extrema, we need to evaluate the
function at these critical numbers, but also at the end points of negative one and four.
So the largest function value of these four will be the absolute maximum, and the least
function value will be the absolute minimum. And to save some time, we're going to go ahead and
evaluate the function using the calculator. I've already typed the function into Y one, so we'll
determine these function values from the home screen. We'll press VARS, right arrow, enter,
enter, and now we can just enter the X values in parentheses. So in parentheses, we have zero,
right arrow, enter, enter, then we have three, we also have negative one and four.
So notice of these four function values, the largest function value is one point two five or
one and one fourth. That's our absolute maximum at X equals negative one, and the least function
value is negative six point seven five or negative six and three fourths at X equals three. This
will be our absolute minimum. So let's go ahead and record these function values.
So again, the absolute maximum is one point two five or one and one fourth at
X equals negative one, and the absolute minimum is negative six point seven five or
negative six and three fourths at X equals three. Let's go ahead and verify this graphically.
Notice how I've graphed this function only on the interval from negative one to positive four. The
largest function value, which occurs here, is one point two five at X equals negative one, which is
the absolute maximum, and then we had an absolute minimum of negative six point seven five at X
equals three located here. So notice how the absolute maximum occurred at an endpoint and
the absolute minimum occurred at a critical number. This is why we have to determine the
function values at both the critical numbers and the end points. I hope you found this helpful.