We're given an f double prime of x
equals four x plus three,
and f prime of negative three equals four
and f of negative three equals three.
We're asked find f prime of x and f of three.
Since we can think of integration
as the opposite operation of differentiation,
if we integrate the second derivative,
we recover the first derivative
plus a constant.
And if we integrate the first derivative,
we recover the original function plus a constant.
To begin we will integrate the second derivative
to recover the first derivative,
which gives us the integral of four x plus three.
Which is equal to the integral of four x
or four x to the first,
which is four times x to the second divided by two
plus the integral of three, which is three x plus C.
Let's go ahead and simplify.
Two and four share a common factor of two.
There's one, two, and two, and two twos and four,
which gives us two x squared
plus three x plus C.
And since we integrated the second derivative here,
this is the first derivative plus a constant.
So now we know f prime of x is equal to two x squared
plus three x plus C.
From here since we know f prime of negative three
equals four,
we're going to determine the specific value of C
for our derivative function.
Again, since f prime of negative three equals four,
if we substitute negative three for x
into the derivative function,
the function value must be four
which gives us the equation four equals two times
the square of negative three
plus three times negative three plus C.
And now we solve for C.
We have four equals,
the square negative three is nine.
Two times nine is 18.
You have 18 minus nine plus C.
Simplifying we have four equals nine plus C.
Subtracting nine on both sides,
we have C equals negative five.
So now we know the specific first derivative function
is f prime of x equals two x squared
plus three x,
and since C is negative five, we have minus five.
So this is one of the answers the question asks for.
And now that we have the first derivative,
we can begin recovering the original function
by integrating the first derivative.
So we have the integral of
two x squared plus three x minus five,
which is equal to the integral of two x squared
which is two times x to the third divided by three,
plus the integral of three x,
which is three times x squared divided by two
minus integral of five, which is five x,
and then plus C.
Which means we now know the original function f of x
is equal to two thirds x cubed,
plus the three halves x squared
minus five x plus C.
And now we can find the value of C
because we know f of negative three equals three.
Since f of negative three is equal to three,
if we substitute negative three for x,
the function value must be three
which gives us the equation three equals
two-thirds times the cube of negative three
plus three halves times the square of negative three,
minus five times negative three plus C.
Simplifying, we have three equals.
Let's simplify this right side on the calculator.
On the right side of the equation we have 10.5 plus C.
Let's convert 10.5 to a fraction
by pressing math enter enter, which is 21 halves.
So we have three equals 21 halves plus C,
which means C is equal to three minus 21 halves,
C is equal to negative 15 halves.
Of course we can check this on the calculator if we need to,
three minus 21 halves
is negative 15 halves.
So now we know the original function f of x
is f of x equals
two-thirds x cubed
plus three halves x squared,
minus five x.
And since C is negative 15 halves, we have minus 15 halves.
So while we're not giving f of x as an answer,
we do need to find f of x
in order to determine in the function value f of three.
F of three
is equal to
two-thirds times the cube of three
plus three halves times the square of three
minus five times three, minus 15 halves.
Now we'll go back to the calculator one more time.
F of three is equal to nine.
I hope you found this helpful.