Question

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

Explanation:

Step1: Find the derivative of the function

The derivative of \(f(x)=(x - 4)^{3}-2\) using the chain rule \((u^{n})^\prime=nu^{n - 1}u^\prime\) (where \(u=x - 4\), \(n = 3\)).
\(f^\prime(x)=3(x - 4)^{2}\times1=3(x - 4)^{2}\)

Step2: Determine where the derivative is positive

Since \((x-4)^{2}\geq0\) for all real \(x\), and \(3>0\), then \(f^\prime(x)=3(x - 4)^{2}\geq0\) for all \(x\in(-\infty,\infty)\). The equality holds when \(x = 4\) (where \(f^\prime(x)=0\)), but the function \(y = f(x)\) is still non - decreasing (and in fact, since the function is a cubic function which is a smooth curve, it is increasing on the entire real line).

Answer:

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