Question

The function f(x) = (x²− 1) |x²− 3x + 2| + cos (|x|) is NOT differentiable at

• A. −1
• B. 0
• C. 1
• D. 2

Answer 1

Answer: (D)

2

Answer 2

We have |x| = $\left{ \begin{matrix}

• xif & x < 0 \ xif & x \geq 0 \end{matrix} \right.\ $

⇒ |x² − 3x + 2| = |(x − 1)(x − 2)|

=$\left\{ \begin{matrix} (x - 1)(x - 2)if \\ (x - 1)(2 - x)if \\ (x - 1)(x - 2)if \end{matrix}\begin{matrix} x \leq 1 \\ 1 < x \leq 2 \\ x \geq 0 \end{matrix} \right.\$

As cos (−θ) = cos θ ⇒ cos |x| = cos x

∴ Given function can be written as

∴ $\left{ \begin{matrix} \left( x^{2} - 1 \right)(x - 1)(x - 2) + \cos xif \

• \left( x^{2} - 1 \right)(x - 1)(x - 2) + \cos xif \ \left( x^{2} - 1 \right)(x - 1)(x - 2) + \cos xif \end{matrix}\begin{matrix} x \leq 1 \ 1 < x \leq 2 \ x > 2 \end{matrix} \right.\ $

This function is differentiable at all points except possibly at x = 1 and x = 2.

Lf’(1) = $\left\{ \frac{d}{dx}\left\lbrack \left( x^{2} - 1 \right)(x - 1)(x - 2) + \cos x \right\rbrack \right\}_{x = 1}$= − sin 1

Rf’(1) = $\left\{ \frac{d}{dx}\left( - \left( x^{2} - 1 \right)(x - 1)(x - 2) + \cos x \right) \right\}_{x = 1}$= − sin 1

∴ Lf’(1) = Rf’(1) ∴ f is differentiable at x = 1.

$Lf'(2) = \left\{ \frac{d}{dx}\left( - \left( x^{2} - 1 \right)(x - 1)(x - 2) + \cos x \right) \right\}_{x = 2}$

$Rf'(2) = \left\{ \frac{d}{dx}\left( \left( x^{2} - 1 \right)(x - 1)(x - 2) + \cos x \right) \right\}_{x = 2}$

∴ Lf’(2) ≠ Rf’(2) ∴ f is not differentiable at x = 2.