Question

Let $f\left(x\right)=\{\begin{array}{cc}-2\sin x& \text{if}x\le \frac{-\pi }{2}\\ A\sin x+B& \text{if}\frac{-\pi }{2}<x<\frac{\pi }{2}\\ \cos x& \text{if}x\ge \frac{\pi }{2}\end{array}$

For what values of $A$ and $B$, the function $f\left(x\right)$ is continuous throughout the real line ?

• A. $A=1,B=1$
• B. $A=-1,B=1$
• C. $A=-1,B=-1$
• D. $A=1,B=-1$

Answer 1

The correct option is B $A=-1,B=1$

Continuity at $x=-\frac{\pi }{2}$
$-2\times (-1)=-A+B\cdots \left(1\right)$

Continuity at $x=\frac{\pi }{2}$
$0=A+B\cdots \left(2\right)$
Using $\left(1\right)\&\left(2\right)$, we get
$A=-1,B=1$