Question:
The domain of derivative of real valued function
\[
f(x)=(x^2-x-2)|x^2+x-6|
\]
is
The domain of derivative of real valued function
\[
f(x)=(x^2-x-2)|x^2+x-6|
\]
is
Show Hint
For expressions containing modulus, first locate zeros of the expression inside modulus. Those are the first points to test differentiability.
Updated On: Jun 15, 2026
- \(\mathbb R\)
- \(\mathbb R-\{-3\}\)
- \(\mathbb R-\{-3,2\}\)
- \(\mathbb R-\{-3,-1,2\}\)
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The Correct Option is C
Solution and Explanation
Concept:
Derivative of function involving modulus fails where expression inside modulus becomes zero and changes sign.
Step 1: Factor modulus expression.
Inside modulus \[ x^2+x-6 \] Factorize \[ =(x+3)(x-2) \] Zeros occur at \[ x=-3,\qquad x=2 \]
Step 2: Analyze differentiability.
Absolute value function changes sign at roots. Derivative fails at points where modulus changes sign. Thus derivative undefined at \[ x=-3,\qquad x=2 \]
Step 3: Write domain.
Hence derivative exists everywhere except these points. \[ Domain= \mathbb R-\{-3,2\} \] Therefore \[ \boxed{\mathbb R-\{-3,2\}} \]
Step 1: Factor modulus expression.
Inside modulus \[ x^2+x-6 \] Factorize \[ =(x+3)(x-2) \] Zeros occur at \[ x=-3,\qquad x=2 \]
Step 2: Analyze differentiability.
Absolute value function changes sign at roots. Derivative fails at points where modulus changes sign. Thus derivative undefined at \[ x=-3,\qquad x=2 \]
Step 3: Write domain.
Hence derivative exists everywhere except these points. \[ Domain= \mathbb R-\{-3,2\} \] Therefore \[ \boxed{\mathbb R-\{-3,2\}} \]
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