We want to find the points
where the tangent line
to the graph of f of x equals two divided by x
is parallel to the line y equals negative 8/3 x plus four.
Remember if lines are parallel, their slopes are equal.
Which means you're trying to find the points
on the graph of f of x equals two divided by x,
where the slope is equal to negative 8/3.
Which means we'll have to set
the derivative of f of x equal to negative 8/3,
and solve for x.
This will give us the x coordinates of the points
and then we'll have to evaluate the function of the x values
to determine the y coordinates.
So we are given f of x
equals two divided by x,
we want to rewrite this using a negative exponent
before we find the derivative.
Two divided by x is equivalent to
two divided by x to the first,
which is equal to two x, the power of negative one.
And now, it's find the first derivative,
f prime of x is equal to the derivative of two x,
the power of negative one with respect to x,
which equal to two times negative one,
x the power of negative one minus one,
which is equal to negative two x the power of negative two,
which is equivalent to negative two divided by x-squared.
And now again, because we're looking for the points
where the tangent line is parallel to
y equal was negative 8/3 x plus three.
We will now set the first derivative,
equal to negative 8/3.
So if f prime of x must equal negative 8/3,
then we know that negative two divided by x-squared
must equal negative 8/3.
Now this all for x, we across multiply
or it can multiply both sides of the equation
by the least common denominator, which is three x-squared.
I'm going to go ahead and multiply both sides
of the equation by three x-squared.
Simplifying on the left, x-squared divided by x-squared,
simplifies the one.
We're left with of negative six equals
on the right three divided by three simplifies to one.
On the right, We have negative eight x-squared.
And now let's isolate x-squared
by dividing both sides by negative eight.
Simplifying, we have x-squared equals
negative six divided by of eight,
simplifies to 3/4.
And now to solve for x,
we take the square root of both sides of the equation.
Remember, we are going to have two solutions.
We include a plus or minus on the left,
which gives us x equals plus or minus
the square root of 3/4, is equal to the square root of three
divided by the square root of four.
And since a square root of four is equal to two.
This simplifies a plus or minus a square root of three
divided by two.
Notice how we have two x values here.
So, there are going to be two points
Where the tangent line to the graph of f of x
is parallel to the given line.
Where one point has an x coordinate
of square root of three divided by two.
And the second point has an x coordinate
of negative square root of three divided by two.
Now, need to work on determining the y coordinates
by evaluating the function at these x values.
Let's first find f of square root of three divided by two,
which is equal to two divided by
square root of three divided by two.
Remember dividing by a fraction is equivalent
to multiplying by the reciprocal,
and therefore two divided by the square root of three
over two is equal to two.
Or if we want two over one times
the reciprocal of square root of three divided by two,
which is two divided by square root of three,
which gives us four divided by square root of three,
which is perfectly fine.
This is an exact value,
but I also want to show how to rationalize the denominator,
just in case you're asked to do so.
To rationalize the denominator
of four divided by square root of three,
we multiply the numerator and denominator
by the square root of three,
Which gives us four square root of three
divided by square root of three times square three is three.
So we can express the y coordinate
as either four divided by square root of three,
or four square root of three divided by three.
I'm going to go and just use four divided by square three.
And now we need to find
f of negative square three divided by two.
Well, the only difference here is the sign.
We have two divided by
negative square root of three divided by two,
which is just going to be
negative four divided by square root of three,
or negative four square root of three divided by three.
And again, I'm going to go and just use
negative four divided by square root of three.
So these are the two points,
Where the tangent lines to f of x are parallel to the line
y equals negative a 8/3 x plus four.
So let's take a look at this graphically.
The function is graphed in blue.
The given line y equals negative 8/3 x plus four
is the green line.
And we just found these two red points on the function,
here and here, where the tangent lines are parallel
to the given green line.
I hope you found this helpful.