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## Homework Statement

Let f be a differentiable function, defined for all real numbers x, with the following properties:

1. [tex]f'(x) = ax^2 + bx[/tex]

2. [tex]f'(1) = 6[/tex] and [tex]f"(1) = 18[/tex]

3. [tex]\int_{1}^{2} f(x)dx = 18[/tex]

Find f(x).

## Homework Equations

## The Attempt at a Solution

Using the first two properties, I did some algebra (solve the second derivative equation for b and write the first derivative equation in terms of a and solve), and found that a = 12 and b = -6. Using this I took the intergral of the first derivative and got this:

[tex]f(x) = 4x^3 - 3x^2 + c[/tex]

The problem is that this equation doesn't satisfy the 3rd property. Is what I have so far correct? If not, how can I account for that 3rd property when I'm solving for a/b and finding f(x)? Thanks!