Question

F(x)=x^3−9x

Answer 1

Option B

Over the [-4,-1], the function has positive rate of change

Solution:

The average rate of change is given by formula:

$Rate\ of\ change = \frac{f(b)-f(a)}{b-a}$

Given function is:

$f(x) = x^3-9x$

Option A

[-2,1]

Find f(-2) and f(1)

Substitute x = -2 in given function

$f(-2) = (-2)^3 -9(-2)\\\\f(-2) = -8 + 18 = 10$

Substitute x = 1 in given function

$f(1) = 1^3-9(1) = 1-9 = -8$

Find rate of change:

$Rate\ of\ change = \frac{f(-2)-f(1)}{-2-1}\\\\Rate\ of\ change = \frac{10-(-8)}{-3} = \frac{18}{-3} = -6$

Thus this has a negative rate of change

Option B

[-4,-1]

Find f(-4) and f(-1)

Substitute x = -4 in given function

$f(-4) = (-4)^3 -9 (-4) = -64 + 36 = -28$

Substitute x = -1 in given function

$f(-1) = (-1)^3 -9(-1)\\\\f(-1) = -1 + 9 = 8$

Find rate of change:

$Rate\ of\ change = \frac{8-(-28)}{-1-(-4)} = \frac{36}{3} = 12$

Thus this has a positive rate of change

Option C

[-1,2]

Substitute x = -1 in given function

$f(-1) = (-1)^3 -9(-1)\\\\f(-1) = -1 + 9 = 8$

Substitute x = 2 in given function

$f(2) = 2^3 -9(2) = 8-18 = -10$

Find rate of change:

$Rate\ of\ change = \frac{-10-8}{2-(-1)} = \frac{-18}{3} = -6$

Thus this has a negative rate of change

Option D

[-3,3]

Substitute x = -3 in given functiom

$f(-3) = (-3)^3 - 9(-3) = -27 + 27 = 0$

Substitute x = 3 in given function

$f(3) = 3^3 -9(3) = 27 - 27 = 0$

Find rate of change:

$Rate\ of\ change = \frac{0-0}{3-(-3)} = 0$

Thus rate of change is 0