Find f(x) given properties of the derivatives.

  • Thread starter Skizye
  • Start date
  • #1
5
0

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!
 

Answers and Replies

  • #2
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.
 
  • #3
34,900
6,644

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.
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!
 

Related Threads on Find f(x) given properties of the derivatives.

Replies
11
Views
2K
Replies
1
Views
1K
Replies
5
Views
3K
Replies
6
Views
405
Replies
0
Views
1K
  • Last Post
Replies
3
Views
690
Replies
2
Views
2K
  • Last Post
Replies
2
Views
6K
Top