using only the values given in the table for the function, $f(x)=x^{3}-3x - 2$, what is the interval of…

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

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

Answer

Explanation:

Step1: Understand the concept of a decreasing function

A function (y = f(x)) is decreasing on an interval if for any two values (x_1) and (x_2) in the interval with (x_1<x_2), we have (f(x_1)>f(x_2)).

Step2: Check each interval

  • For the interval ((-4,1))
    • When (x=-4), (f(-4)=-54); when (x = - 3), (f(-3)=-20). Since (-54<-20), the function is not decreasing on ((-4,1)) as a whole.
  • For the interval ((-4,-1))
    • When (x=-4), (f(-4)=-54); when (x=-3), (f(-3)=-20). Since (-54 < - 20), the function is not decreasing on ((-4,-1)) as a whole.
  • For the interval ((-1,1))
    • When (x=-1), (f(-1) = 0); when (x=0), (f(0)=-2); when (x = 1), (f(1)=-4).
    • Since (0>-2) and (-2>-4) (i.e., for (x_1=-1,x_2 = 0) with (-1<0) and (f(-1)>f(0)) and for (x_1=0,x_2 = 1) with (0<1) and (f(0)>f(1))), the function is decreasing on ((-1,1)).
  • For the interval ((-1,2))
    • When (x = 1), (f(1)=-4); when (x=2), (f(2)=0). Since (-4<0), the function is not decreasing on ((-1,2)) as a whole.

Answer:

((-1,1))