Question

Let $f'(x) = 3(x^2 + 2x) - \frac{4}{x^3} + 5$, $f(1) = 0$. Then, $f(x)$ is:

• A. $x^3 + 3x^2 + \frac{2}{x^2} + 5x + 11$
• B. $x^3 + 3x^2 + \frac{2}{x^2} + 5x - 11$
• C. $x^3 + 3x^2 - \frac{2}{x^2} + 5x - 11$
• D. $x^3 - 3x^2 - \frac{2}{x^2} + 5x - 11$

Hint:

To find the function from its derivative, integrate term by term, and use initial conditions to find the constant of integration.

Answer 1

The Correct Option is C

We are given that $f'(x) = 3(x^2 + 2x) - \frac{4}{x^3} + 5$. To find $f(x)$, we need to integrate $f'(x)$ with respect to $x$. First, break it into parts: \[ f'(x) = 3(x^2 + 2x) - \frac{4}{x^3} + 5. \] Now integrate each term separately: - The integral of $3(x^2 + 2x)$ is $x^3 + 3x^2$. - The integral of $-\frac{4}{x^3}$ is $\frac{2}{x^2}$. - The integral of 5 is $5x$. Thus, we have: \[ f(x) = x^3 + 3x^2 - \frac{2}{x^2} + 5x + C. \] We are given that $f(1) = 0$. Substituting $x = 1$: \[ 0 = 1^3 + 3(1^2) - \frac{2}{1^2} + 5(1) + C = 1 + 3 - 2 + 5 + C. \] Simplifying: \[ 0 = 7 + C \quad \Rightarrow \quad C = -7. \] Thus, the function is: \[ f(x) = x^3 + 3x^2 - \frac{2}{x^2} + 5x - 7. \] The closest option is (3).