Question

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

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

Answer 1

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`