Question:
The function \( f(x) = (x^2 - 1)|x^2 - 3x + 2| + \cos|x| \) is non-differentiable at
The function \( f(x) = (x^2 - 1)|x^2 - 3x + 2| + \cos|x| \) is non-differentiable at
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If modulus is multiplied by zero at a point, cusp may vanish — check carefully.
Updated On: Apr 23, 2026
- \( -1 \)
- \( 0 \)
- \( 1 \)
- \( 2 \)
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The Correct Option is D
Solution and Explanation
Concept:
Non-differentiability occurs where modulus argument is zero or function is non-smooth.
Step 1: Solve inside modulus. \[ x^2 - 3x + 2 = 0 \Rightarrow x = 1, 2 \]
Step 2: Check smoothness: At \( x = 1 \): \[ (x^2 - 1) = 0 \Rightarrow \text{smooth} \] At \( x = 2 \): \[ (x^2 - 1) \neq 0 \Rightarrow \text{kink exists} \]
Step 3: \( \cos|x| \) is differentiable everywhere except possibly at 0, but it is smooth at 0. Final Answer: \[ x = 2 \]
Step 1: Solve inside modulus. \[ x^2 - 3x + 2 = 0 \Rightarrow x = 1, 2 \]
Step 2: Check smoothness: At \( x = 1 \): \[ (x^2 - 1) = 0 \Rightarrow \text{smooth} \] At \( x = 2 \): \[ (x^2 - 1) \neq 0 \Rightarrow \text{kink exists} \]
Step 3: \( \cos|x| \) is differentiable everywhere except possibly at 0, but it is smooth at 0. Final Answer: \[ x = 2 \]
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