(a) Check the differentiability of f ( x )=| cos x | at x =(π)/2.

$$\begin{aligned} & \mathrm{f}(\mathrm{x})=|\cos \mathrm{x}|=\left\{\begin{array}{cc} \cos \mathrm{x} & 0 \leq \mathrm{x} \leq \frac{\pi}{2} \\ -\cos \mathrm{x} & \frac{\pi}{2} \leq \mathrm{x} \leq \pi \end{array}\right. \\ & \text { LHD at } \frac{\pi}{2}=\lim _{h \rightarrow 0} \frac{\mathrm{f}\left(\dfrac{\pi}{2}-h\right)-\mathrm{f}\left(\dfrac{\pi}{2}\right)}{-h}=\lim _{h \rightarrow 0} \frac{\cos \left(\dfrac{\pi}{2}-h\right)-0}{-h}=\lim _{h \rightarrow 0} \frac{\sin h}{-h}=-1 \end{aligned} $$ $$\begin{aligned} & \frac{d y}{d \mathrm{x}}=2 A \cos 2 \mathrm{x}-2 B \sin 2 \mathrm{x} \\ & \Rightarrow \frac{d^{2} y}{d \mathrm{x}^{2}}=-4 A \sin 2 \mathrm{x}-4 B \cos 2 \mathrm{x}=-4 y \\ & \Rightarrow \frac{d^{2} y}{d \mathrm{x}^{2}}+4 y=0 \\ & \Rightarrow k=-4 \end{aligned} $$