f(x) = 24 / x^3 and f(2) = 12. f(-1) =

Answer:

0

Explanation:

Step1: Integrate $f'(x)$

First, rewrite $f'(x)=\frac{24}{x^{3}} = 24x^{-3}$. Integrating using the power - rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $f(x)=\int24x^{-3}dx=24\times\frac{x^{-3 + 1}}{-3 + 1}+C=-12x^{-2}+C=-\frac{12}{x^{2}}+C$.

Step2: Find the value of $C$

Given $f(2) = 12$, substitute $x = 2$ into $f(x)$: $12=-\frac{12}{2^{2}}+C$. Simplify the right - hand side: $12=-3 + C$. Solving for $C$, we get $C = 15$. So, $f(x)=-\frac{12}{x^{2}}+15$.

Step3: Calculate $f(-1)$

Substitute $x=-1$ into $f(x)$: $f(-1)=-\frac{12}{(-1)^{2}}+15=-12 + 15=3$.