question_answer

Find the values of a and b so that the function \[f(x)=\left\{ \begin{matrix}    x+a\sqrt{2}\sin x, & 0\underline{<}\,\,x<\pi /4  \\    2x\,\,\cot x+b, & \pi /4\,\,\underline{<}x\,\,\underline{<}\,\,\pi /2  \\    a\,\,\cos 2x-b\sin x, & \pi /2<\,\,x\,\,\underline{<}\,\pi   \\ \end{matrix} \right.\] is continuous for \[0\underline{<}\,\,x<\pi \text{.}\]

A)  \[\frac{\pi }{6},\]\[-\frac{\pi }{6}\]                     

B)  \[\frac{\pi }{6},\,\,\,-\frac{\pi }{12}\]

C)  \[\frac{\pi }{4},\]\[-\frac{\pi }{12}\]                   

D)  \[\frac{\pi }{4},\]\[-\frac{\pi }{6}\]

E)  None of these