Question

The function \( f(x) = x^2(x-2) \) is strictly decreasing in

• A. \( (1,2) \)
• B. \( (-1,1) \)
• C. \( \left(\frac{4}{3},\infty\right) \)
• D. \( (-1,0) \)
• E. \( \left(0,\frac{4}{3}\right) \)

Hint:

For increasing/decreasing, analyze sign of derivative using intervals.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
A function is strictly decreasing on an interval where its first derivative is negative (\(f'(x)<0\)). To find these intervals, we first need to find the critical points of the function, which are the points where the derivative is zero or undefined.
Step 2: Key Formula or Approach:
1. Find the first derivative, \(f'(x)\). 2. Find the critical points by solving \(f'(x) = 0\). 3. Create a sign chart for \(f'(x)\) using the critical points to determine the intervals where \(f'(x)\) is positive (increasing) and negative (decreasing). 4. Identify the interval where \(f'(x)<0\).
Step 3: Detailed Explanation:
1. Find the first derivative. First, expand the function for easier differentiation: \[ f(x) = x^2(x-2) = x^3 - 2x^2 \] Now, differentiate with respect to x: \[ f'(x) = 3x^2 - 4x \] 2. Find the critical points. Set the derivative equal to zero: \[ 3x^2 - 4x = 0 \] Factor out x: \[ x(3x - 4) = 0 \] The critical points are \(x = 0\) and \(3x - 4 = 0 \implies x = \frac{4}{3}\). 3. Create a sign chart for f'(x). The critical points divide the number line into three intervals: \((-\infty, 0)\), \((0, 4/3)\), and \((4/3, \infty)\). We test a point from each interval to see the sign of \(f'(x)\).
Interval \((-\infty, 0)\): Let's test \(x = -1\). \(f'(-1) = 3(-1)^2 - 4(-1) = 3 + 4 = 7\) (Positive, so f is increasing).
Interval \((0, 4/3)\): Let's test \(x = 1\). \(f'(1) = 3(1)^2 - 4(1) = 3 - 4 = -1\) (Negative, so f is strictly decreasing).
Interval \((4/3, \infty)\): Let's test \(x = 2\). \(f'(2) = 3(2)^2 - 4(2) = 12 - 8 = 4\) (Positive, so f is increasing).
4. Identify the decreasing interval. The function is strictly decreasing where \(f'(x)<0\), which is the interval \((0, 4/3)\).
Step 4: Final Answer:
The function is strictly decreasing in the interval \((0, \frac{4}{3})\).