Question

let (f(x)=-2x(x - 1)). then (f(0)=) and after simplifying (f(x)=) hint: you may want to expand and simplify the expression for (f(x)) first.

Explanation:

Step1: Expand f(x)

\[

$$\begin{align*} f(x)&=-2x(x - 1)\\ &=-2x^2+2x \end{align*}$$

\]

Step2: Differentiate f(x) using power - rule

The power - rule states that if \(y = ax^n\), then \(y^\prime=anx^{n - 1}\).
For \(y=-2x^2+2x\), \(f^\prime(x)=-2\times2x^{2 - 1}+2\times1x^{1 - 1}=-4x + 2\)

Step3: Find f'(0)

Substitute \(x = 0\) into \(f^\prime(x)\):
\(f^\prime(0)=-4\times0+2=2\)

Answer:

\(f^\prime(0)=2\)
\(f^\prime(x)=-4x + 2\)