Question

(a) find the derivative f(x) of the function f(x) = $\frac{x^{3}}{8}$.
(b) graph f(x) and f(x) side by side using separate sets of coordinate axes.
(c) for what values of x, if any, is f positive? zero? negative?
(d) over what intervals of x - values, if any, does the function y = f(x) increase as x increases? decrease as x increases? how is this related to the findings in part (c)?
(a) f(x) = □

Explanation:

Step1: Recall power - rule for derivatives

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$, where $a$ is a constant and $n$ is a real number. For the function $f(x)=\frac{x^{3}}{8}=\frac{1}{8}x^{3}$, here $a = \frac{1}{8}$ and $n = 3$.

Step2: Apply the power - rule

$f^\prime(x)=\frac{1}{8}\times3x^{3 - 1}=\frac{3}{8}x^{2}$

Answer:

$\frac{3}{8}x^{2}$