Question

Let $f\left(x\right)=\{\begin{array}{c}-2\sin xfor-\pi \le x\le -\frac{\pi }{2}\\ a\sin x+bfor-\frac{\pi }{2}<x<\frac{\pi }{2}\\ \cos xfor\frac{\pi }{2}\le x<\pi \end{array}$ , If f is continuous on $[-\pi ,\pi ]$ then find the values of a & b.

• A. $a=-1,b=1$
• B. $a=0,b=1$
• C. $a=1,b=0$
• D. $a=-1,b=-1$

Answer 1

The correct option is A $a=-1,b=1$

$f\left(x\right)=\{\begin{array}{c}-2\sin xfor-\pi \le x\le -\frac{\pi }{2}\\ a\sin x+bfor-\frac{\pi }{2}<x<\frac{\pi }{2}\\ \cos xfor\frac{\pi }{2}\le x<\pi \end{array}$
For $f\left(x\right)$ to be continuous on $[-\pi ,\pi ]$ limit at point $x=-\frac{\pi }{2},\frac{\pi }{2}$ should exist.
$\Rightarrow b-a=2$ and $a+b=0$
$\therefore a=-1,b=1$