Question

1. consider the function ( f ( x ) = \frac { ( x - 4 ) ^ { 3 } } { 2 } - 5 ) drawn below. what are the regions of the domain where ( f ( x ) ) is increasing? write the regions in interval notation separated by commas. ( ( square, 4 ), ( square, square ) )

Explanation:

Step1: Find the derivative of the function

The function is \( f(x)=\frac{(x - 4)^{3}}{2}-5 \).
Using the power rule \((x^n)^\prime=nx^{n - 1}\), the derivative \(f^\prime(x)=\frac{3(x - 4)^{2}}{2}\).
Since \((x-4)^{2}\geq0\) for all real \(x\), and \(\frac{3}{2}>0\), then \(f^\prime(x)=\frac{3(x - 4)^{2}}{2}\geq0\) for all \(x\in R\).

Step2: Analyze the increasing - decreasing nature

A function \(y = f(x)\) is increasing when \(f^\prime(x)>0\) and has a horizontal tangent when \(f^\prime(x) = 0\).
The derivative \(f^\prime(x)=\frac{3(x - 4)^{2}}{2}=0\) when \(x = 4\).
For \(x
eq4\), \(f^\prime(x)>0\).

Answer:

\((-\infty,4),(4,\infty)\)