Question

If `f''(t)=2+2t` and `f(0)=2` and `f(3)=2` then find `f(t)`?

Answer 1

`f(t) = t^2 + t^3/3 - 6t + 2`

Explanation:

If:

`f''(t)=2+2t`

Then integrating wrt `t` we get:

`f'(t)=2t + 2t^2/2 + a`
`\ \ \ \ \ \ \ \ =2t + t^2 + a`

Then integrating wrt `t` again, we get:

`f(t) = 2t^2/2 + t^3/3 + at + b`
`\ \ \ \ \ \ = t^2 + t^3/3 + at + b`

We are given that:

`f(0) = 2 => 0 + 0 + 0a + b = 2 => b=2`
`f(3) = 2 => 9 + 9 + 3a + 2 = 2 => a=-6`

Giving us:

`f(t) = t^2 + t^3/3 - 6t + 2`

Verification:

`f(0) = 2`
`f(3) = 9 + 9 -18 + 2 = 2`
`f'(t) = 2t+t^2 - 6`
`f''(t) = 2+2t`