Question

Let $f\left(x\right)=\{\begin{array}{c}-2\sin x,ifx\le -\frac{\pi }{2}\\ A\sin x+B,if-\frac{\pi }{2}<x<\frac{\pi }{2}\\ \cos x,ifx\ge \frac{\pi }{2}\end{array}$. Then

• A. $f$ is discontinuous for all $A$ and $B$
• B. $f$ is discontinuous for all $A=-1$ and $B=1$
• C. $f$ is discontinuous for all $A=1$ and $B=-1$
• D. $f$ is discontinuous for all real values of $A$ and $B$

Answer 1

Answer: (B)

$f$ is discontinuous for all $A=-1$ and $B=1$

for continuity at $x=\frac{\pi }{2}$ and $x=-\frac{\pi }{2}$
$=-A+B$ -------i

$A+B=0$ ------ii

solving equation i and ii we get,
$A=-1$ and $B=1$