Question

graphing the cubic function quick check
consider the function $f(x) = (x - 4)^3 - 2$. in which interval is the graph increasing? (1 point)
$(-infty, infty)$
$(4, infty)$
$(-infty, 4)$
$(0, infty)$

Explanation:

Step1: Recall the derivative of cubic function

The function is \( f(x)=(x - 4)^3-2 \). The derivative of \( y = x^3 \) is \( y'=3x^2\geq0 \) for all \( x \), and it's positive when \( x
eq0 \). For \( f(x)=(x - 4)^3-2 \), using the chain rule, the derivative \( f'(x)=3(x - 4)^2 \).

Step2: Analyze the sign of the derivative

Since \( (x - 4)^2\geq0 \) for all real numbers \( x \), then \( 3(x - 4)^2\geq0 \) for all \( x \). And \( 3(x - 4)^2 = 0 \) only when \( x = 4 \), but for all other \( x \), \( f'(x)>0 \). So the function is increasing on the entire real line, which is the interval \( (-\infty,\infty) \).

Answer:

\( (-\infty, \infty) \)