We're asked to find the x value
where the maximum value of y
equals e to the power of x divided by 3
minus x squared occurs using Newton's method
and the Newton's method calculator provided.
And we're told to round to 10 decimal places.
Newton's method uses tangent lines
to approximate zeros of functions.
And we use the formula shown here
to make the approximations.
In this case though, this problem's a little bit different.
We're looking for the location of the maximum value
of the given function, not where it's equal to zero.
But remember, if the function does have a maximum value,
it will occur at a critical number,
and critical numbers occur
where the derivative is equal to zero or undefined,
which means we will be using y prime
for f of x in Newton's method.
So let's first write y as y equals e to the power of 1/3x
minus x squared.
And again, because we're looking for the critical number
where y has a maximum value,
we will use y prime for f of x
in the Newton's method formula.
Well, the derivative of y
is equal to e to the power of 1/3x times 1/3 minus 2x,
or 1/3 e to the power of 1/3x minus 2x,
and then f prime of x in Newton's method
will be the derivative of f of x
or the derivative of y prime.
And the derivative of 1/3 e to the power of 1/3x minus 2x
is 1/3 e to the power of 1/3x times 1/3 minus 2,
or 1/9 e to the power of 1/3x minus 2.
So again, to better understand what's happening here,
let's look at the graph on the right.
In purple, we have the graph of the original function,
y equals e the power of x divided by 3 minus x squared.
And we're looking for the x value
where this function has a maximum,
which is this location here.
But again, Newton's method approximates zeros of functions.
And this maximum will occur at a critical number
where the derivative function is equal to zero.
And I've graphed the derivative function or y prime,
which we're using as f of x
in the Newton's method formula here in red.
And notice, the zero of f of x or y prime
does give us the location where y has a maximum.
So we're using Newton's method to approximate
where the derivative function has a zero,
which will give us where y has a maximum.
The initial approximation can be labeled x sub zero
or x sub one.
Let's go and use x sub zero.
And then using the graph,
let's make the initial approximation of 0.5.
Notice, 0.5 is close to where f of x or y prime
is equal to zero.
And then using Newton's method
we will get closer and closer to the real zero
of the derivative function or f of x,
which again will give us a location where y has a maximum.
And again, in this problem we'll be using
the Newton's method calculator provided in more.
We won't be performing the calculations by hand.
So now going to the Newton's method calculator
we begin by entering f of x,
which again is 1/3 e of the power of 1/3x minus 2x.
f prime of x is 1/9 e to the power of 1/3x minus 2.
We're using 0.5 as the initial approximation.
Now, because we're looking for the answer
to 10 decimal places,
we are go going to run the calculator
until we get the same approximation
rounded to 10 decimal places.
So let's let N equals six
meaning we will run six iterations of Newton's method.
And then we scroll down and click submit.
And if we scroll down again,
we have our six approximations and those last four
are the same all the way out
to all the decimal places provided.
And again, we're asked to round to 10 decimal places.
So notice the zero here is in the 10th decimal place
because we have a four to the right.
We round down to 0.1767830910.
Let's go ahead and record this.
So the location where y equals either the power of x
divided by 3 minus x squared
has a maximum value to 10 decimal places
is x is approximately 0.1767830910.
Before we go, let's take a look at this graphically.
Now the graph is only going to show our solution
to three decimal places,
but we use Newton's method to approximate the zero
of the derivative function, which is this point here
the x intercept of the red curve
which is the derivative of the given function
graft in purple
and notice how this does give us a location
where the original function
does have a maximum function value
which we can see here is approximately 1.029.
But again, our question was to find the location
which we can see here to three decimal places
is approximately 0.177.
I hope you found this helpful.