Question:
The function \( f(x) = x^2(x-2) \) is strictly decreasing in
The function \( f(x) = x^2(x-2) \) is strictly decreasing in
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For increasing/decreasing, analyze sign of derivative using intervals.
Updated On: Apr 21, 2026
- \( (1,2) \)
- \( (-1,1) \)
- \( \left(\frac{4}{3},\infty\right) \)
- \( (-1,0) \)
- \( \left(0,\frac{4}{3}\right) \)
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The Correct Option is
Solution and Explanation
Concept:
Function is decreasing where \( f'(x)<0 \).
Step 1: Differentiate.
\[ f'(x) = 2x(x-2) + x^2 = 3x^2 - 4x \]
Step 2: Solve inequality.
\[ 3x^2 - 4x<0 \Rightarrow x(3x-4)<0 \]
Step 3: Find interval.
\[ 0<x<\frac{4}{3} \]
Step 1: Differentiate.
\[ f'(x) = 2x(x-2) + x^2 = 3x^2 - 4x \]
Step 2: Solve inequality.
\[ 3x^2 - 4x<0 \Rightarrow x(3x-4)<0 \]
Step 3: Find interval.
\[ 0<x<\frac{4}{3} \]
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