Question

f(x)=x^{3}-4.5x^{2}-12x + 1

a) find the first and second derivatives.
f(x)=

f(x)=

b) identify the graph that displays f in blue and f in red. ?

Explanation:

Step1: Find the first derivative

Use the power rule \((x^n)^\prime = nx^{n - 1}\).
For \(f(x)=x^{3}-4.5x^{2}-12x + 1\),
\(f^\prime(x)=\frac{d}{dx}(x^{3})-\frac{d}{dx}(4.5x^{2})-\frac{d}{dx}(12x)+\frac{d}{dx}(1)\)
\(f^\prime(x)=3x^{2}-4.5\times2x-12\)
\(f^\prime(x)=3x^{2}-9x - 12\)

Step2: Find the second derivative

Differentiate \(f^\prime(x)=3x^{2}-9x - 12\) with respect to \(x\) using the power rule.
\(f^{\prime\prime}(x)=\frac{d}{dx}(3x^{2})-\frac{d}{dx}(9x)-\frac{d}{dx}(12)\)
\(f^{\prime\prime}(x)=3\times2x-9\)
\(f^{\prime\prime}(x)=6x - 9\)

Answer:

\(f^\prime(x)=3x^{2}-9x - 12\)
\(f^{\prime\prime}(x)=6x - 9\)