If the function $f(x) = \left\{ \begin{array}{l}1 + \sin \frac{{\pi x}}{2}\,\,,\,{\rm{\,\,for}}\,\, - \infty < x \le 1\\\,\,\,\,\,\,\,\,ax + b,\,{\rm{\,\,for}}\,\,1 < x < 3\\\,\,\,\,6\tan \frac{{x\pi }}{{12}},\,{\rm{\,\,for\,\,}}3 \le x < 6\end{array} \right.$ is continuous in the interval $( - \infty ,\,6)$, then the values of $a$ and $b$ are respectively
- A
$0, 2$
- B
$1, 1$
- C
$2, 0$
- D
$2, 1$
✓
Answer
Given function is continuous at all point in $( - \,\infty ,6)$ and at $x = 1,x = 3$ function is continuous.
If function $f(x)$ is continuous at $x = 1,$ then
$\mathop {\lim }\limits_{x \to {1^ - }} \,f(x) = \mathop {\lim }\limits_{x \to {1^ + }} \,f(x)$
$ \Rightarrow 1 + \sin \frac{\pi }{2} = a + b$
$\therefore a + b = 2.....(i)$
If at $x = 3,$ function is continuous, then
$\mathop {\lim }\limits_{x \to {3^ - }} \,f(3) = \mathop {\lim }\limits_{x \to {3^ + }} \,f(x)$
$ \Rightarrow 3a + b = 6\tan \frac{{3\pi }}{{12}}$
$\therefore 3a + b = 6.....(ii)$
From $(i)$ and $(ii),$ $a = 2,b = 0$ .
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