Question

f(x) = x^3 - 9x
over which interval does f have a positive average rate of change?
choose 1 answer:
a -2, 1
b -1, 2
c -3, 3
d -4, -1

Explanation:

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
We need to calculate this for each interval and check which one is positive.

Step 1: Recall the formula for average rate of change

The average rate of change of \( f(x) \) over \([a, b]\) is \( \frac{f(b) - f(a)}{b - a} \). We need to compute this for each option.

Step 2: Analyze Option A: \([-2, 1]\)

First, find \( f(-2) \) and \( f(1) \):
\( f(-2) = (-2)^3 - 9(-2) = -8 + 18 = 10 \)
\( f(1) = (1)^3 - 9(1) = 1 - 9 = -8 \)
Average rate of change: \( \frac{f(1) - f(-2)}{1 - (-2)} = \frac{-8 - 10}{3} = \frac{-18}{3} = -6 \) (Negative)

Step 3: Analyze Option B: \([-1, 2]\)

Find \( f(-1) \) and \( f(2) \):
\( f(-1) = (-1)^3 - 9(-1) = -1 + 9 = 8 \)
\( f(2) = (2)^3 - 9(2) = 8 - 18 = -10 \)
Average rate of change: \( \frac{f(2) - f(-1)}{2 - (-1)} = \frac{-10 - 8}{3} = \frac{-18}{3} = -6 \) (Negative)

Step 4: Analyze Option C: \([-3, 3]\)

Find \( f(-3) \) and \( f(3) \):
\( f(-3) = (-3)^3 - 9(-3) = -27 + 27 = 0 \)
\( f(3) = (3)^3 - 9(3) = 27 - 27 = 0 \)
Average rate of change: \( \frac{f(3) - f(-3)}{3 - (-3)} = \frac{0 - 0}{6} = 0 \) (Zero)

Step 5: Analyze Option D: \([-4, -1]\)

Find \( f(-4) \) and \( f(-1) \):
\( f(-4) = (-4)^3 - 9(-4) = -64 + 36 = -28 \)
\( f(-1) = (-1)^3 - 9(-1) = -1 + 9 = 8 \)
Average rate of change: \( \frac{f(-1) - f(-4)}{-1 - (-4)} = \frac{8 - (-28)}{3} = \frac{36}{3} = 12 \) (Positive)

Answer:

D. \([-4, -1]\)