We're asked to determine the absolute extrema
of the given function over the closed interval
from negative three to positive nine.
Because the given function is continuous
over the closed interval
from negative three to positive nine, we are guaranteed
to have an absolute maximum and an absolute minimum.
To determine the absolute extrema,
we evaluate the given function at the end points,
as well as at the critical numbers.
Let's begin by determining the critical numbers
of the given function, which is where the derivative
is equal to zero or undefined.
If prime of x is equal to the derivative of x of the fourth,
which is four x cubed
minus the derivative of 32x squared, which is 64x
and then plus the derivative of nine,
which is equal to zero.
Notice how the derivative function is a cubic function,
which is never undefined
and therefore to find their critical numbers,
we set the derivative equal to zero and solve for x,
which gives us four x cubed minus 64x equals zero.
Now we will solve this equation by factoring,
looking at the left side, okay, has come in factors four x,
so we factor out four x,
we're left with x squared minus 16.
Of course we can always check this by distributing four x,
we still have four x cubed minus 64x.
And now we can factor x squared minus 16
as a difference of squares, we have four x
and then times two binomial factors,
where the binomial factors are x plus four and x minus four.
So the product on the left is equal to zero,
so the derivative is equal to zero,
when x equals zero or when x equals negative four
or when x equals positive four.
Now I'll check to make sure all of these critical numbers
are on the interval from negative three to positive nine.
Well, zero is in the interval,
but notice how negative four is not in the interval,
so we are not going to evaluate the original function
at x equals negative four.
So we do have to be careful to make sure
the critical numbers are in the given interval
and four is in the interval.
Now we need to find the function values f of negative three,
f of zero, f of four and f of nine.
Let's do this on the next slide.
And now we perform substitution into the original function.
And now let's go to the calculator.
F of negative three is negative 198, f of zero is nine,
f of four is negative 247 and f of nine is 3,978.
Let's go ahead and record these function values.
Notice the least function value is negative 247,
which is the absolute minimum, which occurs at x equals four
and the greatest function value is 3,978,
which is the absolute maximum,
which occurs with x equals nine.
Let's go ahead and record this below.
The function has an absolute minimum value
equal to negative 247 at x equals four
and the function has an absolute maximum value
equal to 3,978 at x equals nine.
And finally before we go, let's verify this graphically.
I've already graphed the given function
over the closed interval
from negative three to positive nine.
Notice the highest point of the function over this interval
is this point here where the function value is 3,978,
this is the absolute maximum at x equals nine
and the lowest point on the graph over this interval
is this point here where the function value or y value
is negative 247, which occurs at x equals four.
The graph verifies our work is correct.
I hope you found this helpful.