Question

let f be the function given by $f(x)=x^{3}-6x^{2}+8x - 2$. what is the instantaneous rate of change of f at $x = 3$?
a -5
b $-\frac{15}{4}$
c -1
d 6
e 17

Explanation:

Step1: Find the derivative of f(x)

The derivative of $f(x)=x^{3}-6x^{2}+8x - 2$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $f'(x)=3x^{2}-12x + 8$.

Step2: Evaluate f'(x) at x = 3

Substitute $x = 3$ into $f'(x)$: $f'(3)=3(3)^{2}-12(3)+8$.
First, calculate $3(3)^{2}=3\times9 = 27$, $12(3)=36$.
Then $f'(3)=27-36 + 8=-1$.

Answer:

C. -1