Question

The domain of derivative of real valued function \[ f(x)=(x^2-x-2)|x^2+x-6| \] is

• A. \(\mathbb R\)
• B. \(\mathbb R-\{-3\}\)
• C. \(\mathbb R-\{-3,2\}\)
• D. \(\mathbb R-\{-3,-1,2\}\)

Hint:

For expressions containing modulus, first locate zeros of the expression inside modulus. Those are the first points to test differentiability.

Answer 1

The Correct Option is C

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 \[ \boxed{\mathbb R-\{-3,2\}} \]