f(x)=-\\frac{6}{x^{4}} and f(1)=-4. f(-1)=
f(x)=-\\frac{6}{x^{4}} and f(1)=-4. f(-1)=
Answer
Explanation:
Step1: Integrate $f'(x)$
First, rewrite $f'(x)=-6x^{-4}$. Integrating using the power - rule $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $\int f'(x)dx=\int-6x^{-4}dx=-6\times\frac{x^{-4 + 1}}{-4+1}+C$. So, $f(x)=2x^{-3}+C$.
Step2: Find the value of $C$
Given $f(1)=-4$, substitute $x = 1$ into $f(x)$: $f(1)=2\times1^{-3}+C$. $-4=2 + C$, then $C=-6$. So, $f(x)=2x^{-3}-6$.
Step3: Calculate $f(-1)$
Substitute $x=-1$ into $f(x)$: $f(-1)=2\times(-1)^{-3}-6$. $f(-1)=2\times(-1)-6=-2 - 6=-8$.
Answer:
$-8$