Find f(x) given properties of the derivatives.

In summary, the given differentiable function has a first derivative of ax^2 + bx, where a = 12 and b = -6. By integrating the first derivative and evaluating it at the limits, the value of c can be determined to satisfy the third property. Therefore, the function f(x) = 4x^3 - 3x^2 + c satisfies all three properties.
  • #1
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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!
 
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  • #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
Skizye said:

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.
Skizye said:
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!
 

1. What are the properties of derivatives?

The properties of derivatives include the power rule, product rule, quotient rule, chain rule, and logarithmic differentiation. These rules help to find the derivative of a function with respect to its independent variable.

2. How do you use the power rule to find the derivative?

The power rule states that the derivative of a function raised to a constant power is equal to the constant multiplied by the function raised to the power minus one. In other words, for a function f(x) = x^n, the derivative is f'(x) = nx^(n-1).

3. Can you explain the product rule?

The product rule states that the derivative of two functions multiplied together is equal to the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function. In other words, for two functions f(x) and g(x), the derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x).

4. How do you find the derivative of a quotient?

The quotient rule states that the derivative of a fraction is equal to the denominator multiplied by the derivative of the numerator, minus the numerator multiplied by the derivative of the denominator, all divided by the denominator squared. In other words, for a function f(x)/g(x), the derivative is (f'(x)g(x) - f(x)g'(x)) / g(x)^2.

5. When should the chain rule be used?

The chain rule is used when the function being differentiated is composed of two or more functions. It states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In other words, for a function f(g(x)), the derivative is f'(g(x)) * g'(x).

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