Question

$f\left(x\right)=(x^2-1)|x^2-3x+2|+cos\left(|x|\right)$ is not differentiable at:

• A. -1
• B. 1
• C. 2

Answer 1

The correct option is C 2

$f\left(x\right)=\{\begin{array}{cc}(x^2-1)(x-1)(x-2))+cosx:& -\infty <x<1\\ -(x^2-1)(x-1)(x-2))+cosx:& 1\le x<2\\ (x^2-1)(x-1)(x-2))+cosx:& x\ge 2\end{array}$
Clearly f(x) is continuous everywhere.
$f^{\prime }\left(x\right)=\{\begin{array}{cc}(4x^3-9x^2+2x+3)-sinx:& x<1\\ -(4x^3-9x^2+2x+3)-sinx:& 1<x<2\\ (4x^3-9x^2+2x+3)-sinx& x>2\end{array}$
At x = 1, Lf' (1) = sin 1, Rf'(1) = sin 1
At x = 2, Lf' (2) = -3-sin2, Rf'(2) = 3- sin2
$\therefore$ f(x) is not differentiable at x =2.