Match each function in List-I with the number of real points at which it is not differentiable (List-II). For the function in (S), consider only the…

A function $|g(x)|$ fails to be differentiable exactly at simple zeros of $g$ (where $g'\ne0$).
The criterion is that the corner appears only where the inner function changes sign linearly. (P) $x^2-4$ has simple zeros at $\pm2$: $2$ points, (3).
(Q) $f(x)=(x-1)(x+1)|x-1|$. At $x=1$ the factor $(x-1)|x-1|$ is differentiable (derivative $0$) and multiplying by the smooth factor $x+1$ keeps it so; elsewhere $f$ is smooth. $0$ points, (1).
(R) The kinks of $|x-1|$ at $1$ and of $|x^2-4|$ at $\pm2$ cannot cancel: $3$ points, (4).
(S) $\cos2x=0$ in $(0,2\pi)$ at $x=\frac\pi4,\frac{3\pi}4,\frac{5\pi}4,\frac{7\pi}4$, all simple zeros: $4$ points, (5).
The correct code is P→3, Q→1, R→4, S→5.