This lesson will show
how to use the second derivative test
to determine any relative extrema
of f of x equals x times e to the power of 2x.
To use the second derivative test,
we'd begin by determining the critical numbers
of the given function,
which is where the first derivative
is equal to zero or undefined.
And if f of x has a critical number
or critical value of x equals c
if f double prime of c is greater than zero or positive,
then f of x is concave up, but x equals c
and therefore, there must be a low point at x equals c,
and therefore, f of c is a relative minimum.
If f double prime of c is less than zero or negative,
then f of x is concave down at x equals c.
And if f of x is concave down at x equals c,
there must be a high point on the graph at x equals c
and therefore, f of c is a relative maximum.
And if f double prime of c equals zero,
then the test is inconclusive
and we'd have to use the first derivative test
or analyze the graph.
Let's begin by determining the critical numbers
of the given function,
which again, means we need to determine
where the first derivative is equal to zero or undefined.
Notice to find the first derivative
we'll need to apply the Product Rule
as well as the Chain Rule.
f prime of x is equal to the first function of x
times your derivative of the second function,
which is the derivative of e to the power of 2x
plus the second function of e to the power of 2x
times the derivative of the first function,
which is derivative of x.
This gives us f prime of x
equals x times the derivative of e to the power of 2x
is e to the power of 2x times your derivative of 2x
which is 2.
e to the power of 2x times 2
is 2 times e to the power of 2x.
And then we have plus e to the power of 2x
times your derivative of x, which is equal to 1.
Simplifying, we have f prime of x equals 2x e to the 2x
plus e to the 2x.
There are no values of X
where the first derivative is undefined,
and therefore, to find any critical numbers
we set the first derivative equal to zero and solve.
Let's first factor out the greatest common factor
of e to the power of 2x, which gives us
e to the power of 2x
times the quantity 2x plus 1 equals zero.
e to the power of 2x is never equal to zero.
The first derivative is only equal to zero,
when 2x plus 1 equals zero.
Solving for x here,
we subtract 1 on both sides and then divide by 2.
We only have one critical number,
which is x equals -1/2.
Let's go ahead and record this above.
And now, we are going to determine the second derivative
so that we can determine the sign of the second derivative
at the critical number of x equals -1/2.
To find the second derivative,
we determine the derivative of the first derivative
in the form of f prime of x equals 2x e to the 2x
plus e to the 2x.
f double prime of x is equal to the derivative
of the first derivative
or the derivative of 2x e to the 2x plus e to the 2x.
And again, notice how we have to apply the Product Rule
to differentiate 2x e to the 2x.
We have f double prime of x
equals the first function of 2x
times the derivative of e to the 2x,
which we already know is e to the 2x times 2
or 2 e to the 2x.
And then plus the second function of e to the 2x
times your derivative of the first function
which is 2x, the derivative of 2x is equal to 2.
And then, plus the derivative of e to the 2x,
which is 2 e to the 2x.
Simplifying, we have f double prime of x
equals 4x e to the 2x.
And then we have plus 2e to the 2x
plus 2 e to the 2x,
which is plus 4 e to the 2x.
And now that we have the second derivative,
we need to determine the sign of the second derivative
at the critical number x equals -1/2.
So f double prime of -1/2
is equal to 4 times -1/2
times e to the power of 2 times -1/2
plus 4 times e to the power of 2 times -1/2.
Simplifying, we have -2 e to the power of -1
plus 4e to the power of -1,
which we can also write as -2 divided by e plus 4
divided by e.
Simplifying, we have 2 divided by e,
which is going to be positive or greater than zero,
which indicates f of x is concave up
at x equals -1/2.
So if the function is concave up at x equals -1/2,
it would look something like this,
which means that x equals -1/2,
we have a low point
and therefore, f of -1/2
would have to be a relative minimum.
So below that state, the relative minimum is f of -1/2.
We'll determine this function value here in just a moment
at x equals -1/2.
So again, the last step here is to determine
the exact relative minimum function value,
but determining f of -1/2.
And we also know we aren't going to have a relative max
so let's go ahead and state none and none.
So again, for our last step
we'll determine the relative minimum function value
by determining f of -1/2.
f of -1/2 is equal to
1/2 times e raised to the power of 2 times -1/2,
which is equal to -1/2 times e to the power of -1,
which we can also write as -1 divided by 2e,
which as a decimal would be approximately -0.184.
But again, we're going to give the exact value
of -1 divided by 2e.
Before we go, let's verify this graphically.
In blue, we have the graph of the original function
f of x equals x times e to the power of 2x.
Notice how the graph does have a low point here
where the x-coordinate is -0.5 or -1/2.
And the y-value, your function value is exactly
1 divided by 2e,
which we can see here is approximately -0.184.
And the graph also verifies the function is concave up
at this low point.
I hope you found this helpful.