First, we know that the function is quadratic. Which means it's a parable. We know that f(0)=0.
So f(x)=ax^2 + bx+ c = ax^2 + bx
Thanks to (a), we know that the function for all x between 0 and 2 is negative. This is explained with the minus sign in front of the integral.
We also know that because f(x) = ax^2 + bx,
- integral from 0 to 2 of f(x) = - 8a/3 - 2b = 4
You just have to do the integral to see it.
Thanks to (b), we know that the function becomes positive between 2 and 3. Now the function is a parable. So a is greater than or equal to 0 and 2 is a solution of f(x)=0.
So f(2) = 4a + 2b = 0
We have a system of equations:
- 8a/3 - 2b = 4
4a + 2b = 0
We can easily find that a = 3 and b = -6
So f(x) = 3x^2 - 6x
Therefore f(5)=3*5^2 - 6*5 = 3*25 - 30 = 75 - 30 = 45