# Find f(x) given properties of the derivatives.

## Homework Statement

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

1. $$f'(x) = ax^2 + bx$$

2. $$f'(1) = 6$$ and $$f"(1) = 18$$

3. $$\int_{1}^{2} f(x)dx = 18$$

Find f(x).

## 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:

$$f(x) = 4x^3 - 3x^2 + c$$

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!

If you integrate f(x) = 4x^3 -3x^2 +c you get x^4 - x^3 + cx +d .

Evaluating the integral at the limits, you will arrive at a value for c that satifies all properties.

Mark44
Mentor

## Homework Statement

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

1. $$f'(x) = ax^2 + bx$$

2. $$f'(1) = 6$$ and $$f"(1) = 18$$

3. $$\int_{1}^{2} f(x)dx = 18$$

Find f(x).

## 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:

$$f(x) = 4x^3 - 3x^2 + c$$

The problem is that this equation doesn't satisfy the 3rd property.
It will if you find the correct value for c.

Your values for a and b are correct. Now, since you know the value of the integral of f, you can solve for c.
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!