We want to determine the critical numbers
of the given function.
The critical numbers are the values of x in the domain
where the first derivative is equal to zero or undefined.
Let's begin by stating the domain of the given function.
We have f of x equals x to the power of negative 3
times natural log x,
which if we wanted to, we could rewrite
as natural log x divided by x cubed.
This is one example where it's helpful to know the domain
of basic functions.
For example, here, the domain of natural log x
is x greater than zero.
And we can quickly verify this by looking at the graph
of the basic natural log function, which I have here,
again, where the domain is x greater than zero,
Notice how we have a vertical asymptote at x equals zero.
So because f of x contains natural log x,
we do have a restriction on the domain
where x is greater than zero.
We also have x cubed in the denominator,
and therefore, x can't equal zero,
otherwise we'd have division by zero.
But notice x equals zero is not greater than zero,
and therefore, this restriction
is already in the given restriction
of x being greater than zero.
So if we wanted to, we could list the domain
using interval notation,
which would be the open interval from zero to infinity.
Now the reason it's important to determine the domain
is that when we go to determine
where the first derivative is equal to zero or undefined,
if the value is not in the domain,
we do not include it as a critical number.
And now the next step is to determine the derivative
and determine where it's equal to zero or undefined.
And we can use either form of the function.
If we use the first form,
we'll need to use the product rule.
If we use the second form,
we'll need to use the quotient rule.
I'm going to go ahead and use the original form
and the product rule.
And therefore, f prime of x
is equal to the first function of x
to the power of negative 3
times derivative of the second function,
which is the derivative of natural log x.
And then we have plus the second function of natural log x
times derivative of the first function,
which is a derivative of x to the power of negative 3.
This gives us f prime of x
equals x the power of negative 3
times the derivative of natural log x,
which is 1 divided by x.
And then we have plus natural log x
times the derivative of x to the power of negative 3,
which is negative 3 times x to the power
of negative 3 minus 1, which is negative 4.
And now let's clean this up.
Remember 1 divided by x
is equal to x to the power of negative 1.
So for this first product, we have x the power of negative 3
times 1 divided by x, or times x to the power of negative 1,
which gives us x the power of negative 4.
And then we have plus negative 3, x to power of negative 4,
natural log x.
Let's just write that as minus 3,
x to the power of negative 4, natural log x.
And again, here, if we want to,
we can write this using positive exponents.
This is equivalent to 1 divided by x to the fourth
minus 3 natural log x divided by x to the fourth.
And now we need to determine
where the derivative is equal to zero or undefined.
Notice how I do have division by zero when x equals zero,
but x equals zero is not in the domain,
and therefore, we don't include x equals zero
as a critical number.
So now we'll set the derivative equal to zero
and solve for x.
This gives us the equation one divided by x to the fourth
minus 3 natural log x
divided by x to the fourth equals zero.
Let's clear the fractions from the equation
by multiplying both sides of the equation
by x to the fourth.
X to the fourth times 1 divided by x to the fourth
is just 1.
And then we have minus x to the fourth
times 3 natural log x divided x to the fourth,
just gives us 3 natural log x.
On the right we have zero.
And now to solve the equation for x,
we will first isolate natural log x.
Let's begin by subtracting 1 on both sides,
which gives us negative 3 natural log x equals negative 1.
And then we'll divide both sides by negative 3.
Simplifying, we have natural log x
is equal to positive 1/3.
And now to solve for x, we'll write the log equation
as an exponential equation.
Remember, natural log x is log base e,
so the equivalent exponential equation
is e to the power of 1/3 equals x.
So again, we have e to the power of 1/3 equals x.
And e to the power of 1/3 is greater than zero,
and therefore, this is the critical number.
And we can list a critical number
as x equals e to the power of 1/3.
Or if want in radical form,
this would be equivalent to the cube root of e.
So again, the critical number of the given function
is x equals, again, e to the power of 1/3,
or if we want, the cube root of e.
So this tells us,
if the given function does have a relative maximum
or relative minimum,
it would have to occur at x equals the cube root of e.
But it doesn't guarantee we will have a relative max or min
at this location.
Before we go, let's look at the graph.
Here we have the graph of the given function.
And the cube root of e is approximately 1.396.
And notice how we do have a high point at this location.
And therefore, the function does have a relative maximum
at x equals e to the power of 1/3,
or x equals the cube root of e.
I hope you found this helpful.