What is f'(x) and f(1)?

Given
f''(x) = 4x+4
and f'(-1)=1 and f(-1) =-2

1 Answer
Jun 8, 2018

Use the antiderivatives to obtain the exact equations for f'(x) and f(x). From that we get:

#f'(x)= 2x^2 + 4x + 3#
and #f(1)=16/3#

Explanation:

We can apply the antiderivative to: #f''(x)=4x+4#
to obtain an equation for the first drivative:

#f'(x)= 2x^2 + 4x + k#

Now let's evaluate #f'(x)#, when #x=-1#, knowing that the result #f'(-1)# is equal to #1#, as stated in the problem:
#f'(-1) = 2*1+4*(-1)+k = -2+k#
#-2+k=1#
#k=3#

So, the exact equation for the first derivative is:
#f'(x)= 2x^2 + 4x + 3#

We repeat the process to obtain an equation to the original function:
#f(x)=2/3x^3+2x^2+3x+k#

And evaluate the function, when #x=-1#, knowing that the result is equal to #-2#:
#f(-1)=2/3*(-1)+2*1+3*(-1)+k=-2/3-1+k#
#-5/3+k=-2#
#k=-1/3#

The exact function then is:
#f(x)=2/3x^3+2x^2+3x-1/3#

FInally, we find:
#f(1)=2/3*1+2*1+3*1-1/3#

#f(1)=16/3#