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