Option B
Over the [-4,-1], the function has positive rate of change
Solution:
The average rate of change is given by formula:
[tex]Rate\ of\ change = \frac{f(b)-f(a)}{b-a}[/tex]
Given function is:
[tex]f(x) = x^3-9x[/tex]
Option A
[-2,1]
Find f(-2) and f(1)
Substitute x = -2 in given function
[tex]f(-2) = (-2)^3 -9(-2)\\\\f(-2) = -8 + 18 = 10[/tex]
Substitute x = 1 in given function
[tex]f(1) = 1^3-9(1) = 1-9 = -8[/tex]
Find rate of change:
[tex]Rate\ of\ change = \frac{f(-2)-f(1)}{-2-1}\\\\Rate\ of\ change = \frac{10-(-8)}{-3} = \frac{18}{-3} = -6[/tex]
Thus this has a negative rate of change
Option B
[-4,-1]
Find f(-4) and f(-1)
Substitute x = -4 in given function
[tex]f(-4) = (-4)^3 -9 (-4) = -64 + 36 = -28[/tex]
Substitute x = -1 in given function
[tex]f(-1) = (-1)^3 -9(-1)\\\\f(-1) = -1 + 9 = 8[/tex]
Find rate of change:
[tex]Rate\ of\ change = \frac{8-(-28)}{-1-(-4)} = \frac{36}{3} = 12[/tex]
Thus this has a positive rate of change
Option C
[-1,2]
Substitute x = -1 in given function
[tex]f(-1) = (-1)^3 -9(-1)\\\\f(-1) = -1 + 9 = 8[/tex]
Substitute x = 2 in given function
[tex]f(2) = 2^3 -9(2) = 8-18 = -10[/tex]
Find rate of change:
[tex]Rate\ of\ change = \frac{-10-8}{2-(-1)} = \frac{-18}{3} = -6[/tex]
Thus this has a negative rate of change
Option D
[-3,3]
Substitute x = -3 in given functiom
[tex]f(-3) = (-3)^3 - 9(-3) = -27 + 27 = 0[/tex]
Substitute x = 3 in given function
[tex]f(3) = 3^3 -9(3) = 27 - 27 = 0[/tex]
Find rate of change:
[tex]Rate\ of\ change = \frac{0-0}{3-(-3)} = 0[/tex]
Thus rate of change is 0
