Question

$f(x) = x^2 + 10$
over which interval does $f$ have a positive average rate of change?
choose 1 answer:
a $-4, -1$
b $-3, 3$
c $-3, 1$
d $-1, 2$

Explanation:

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

Step 1: Recall the function

The function is \( f(x) = x^2 + 10 \).

Step 2: Analyze the intervals

Let's analyze each interval:

Option A: \([-4, -1]\)

• \( a = -4 \), \( b = -1 \)
• \( f(-4) = (-4)^2 + 10 = 16 + 10 = 26 \)
• \( f(-1) = (-1)^2 + 10 = 1 + 10 = 11 \)
• Average Rate of Change: \( \frac{11 - 26}{-1 - (-4)} = \frac{-15}{3} = -5 \) (Negative)

Option B: \([-3, 3]\)

• \( a = -3 \), \( b = 3 \)
• \( f(-3) = (-3)^2 + 10 = 9 + 10 = 19 \)
• \( f(3) = (3)^2 + 10 = 9 + 10 = 19 \)
• Average Rate of Change: \( \frac{19 - 19}{3 - (-3)} = \frac{0}{6} = 0 \) (Zero)

Option C: \([-3, 1]\)

• \( a = -3 \), \( b = 1 \)
• \( f(-3) = (-3)^2 + 10 = 9 + 10 = 19 \)
• \( f(1) = (1)^2 + 10 = 1 + 10 = 11 \)
• Average Rate of Change: \( \frac{11 - 19}{1 - (-3)} = \frac{-8}{4} = -2 \) (Negative)

Option D: \([-1, 2]\)

• \( a = -1 \), \( b = 2 \)
• \( f(-1) = (-1)^2 + 10 = 1 + 10 = 11 \)
• \( f(2) = (2)^2 + 10 = 4 + 10 = 14 \)
• Average Rate of Change: \( \frac{14 - 11}{2 - (-1)} = \frac{3}{3} = 1 \) (Positive)

Answer:

D. \([-1, 2]\)