Question

Number of points where the function f(x) = | x² – 3x + 2 | + cos | x | is not differentiable
1. 3
2. 0
3. 1
4. 2

Answer 1

Correct Answer - Option 4 : 2

∵ f(x) = | x² – 3x + 2 | + cos | x | = |(x – 1)| |(x – 2)| + cos | x |

f(x) = \(\left\{ {\,\begin{array}{*{20}{c}} {{x^2} - 3x + 2 + \cos x,}&{x < 0}&{}&{} \\ {{x^2} - 3x - 2 + \cos x,}&{0 \leqslant x < 1}&{}&{} \\ { - {x^2} + 3x - 2 + \cos x}&{1 \leqslant x < 2}&{}&{} \\ {{x^2} - 3x + 2 + \cos x,}&{x > 2}&{}&{} \end{array}} \right.\)

∴ f' (x) = \(\left\{ {\,\begin{array}{*{20}{c}} {2x - 3 - \sin x,}&{x < 0}&{}&{} \\ {2x - 3 - \sin x,}&{0 \leqslant x < 1}&{}&{} \\ { - 2x + 3 - \sin x,}&{1 \leqslant x < 2}&{}&{} \\ {2x - 3 - \sin x,}&{x > 2}&{}&{} \end{array}} \right.\)

it is clear f(x) is not differentiable at x = 1 and 2.