Question

using only the values given in the table for the function, ( f(x)=x^{3}-3x - 2 ), what is the interval ofvalues over which the function is decreasing?
( (-4,1) )
( (-4,-1) )
( (-1,1) )
( (-1,2) )

Explanation:

Step1: Analyze the function values

A function \(y = f(x)\) is decreasing if when \(x_1

• For the interval \((-4,1)\):
• When \(x=-4\), \(f(-4)=-54\); when \(x = - 3\), \(f(-3)=-20\), \(-54<-20\) (function is increasing in part of this interval).
• For the interval \((-4,-1)\):
• When \(x=-4\), \(f(-4)=-54\); when \(x=-3\), \(f(-3)=-20\), \(-54<-20\) (function is increasing in part of this interval).
• For the interval \((-1,1)\):
• When \(x=-1\), \(f(-1) = 0\); when \(x = 0\), \(f(0)=-2\) (\(0>-2\)); when \(x = 1\), \(f(1)=-4\) (\(-2>-4\)).
• For the interval \((-1,2)\):
• When \(x = 1\), \(f(1)=-4\); when \(x = 2\), \(f(2)=0\), \(-4<0\) (function is increasing in part of this interval).

Answer:

\((-1,1)\)