If the function f(x) = l 1 + x 2 , ,, , , ,for , , - < x 1 , , , … - Vidyadip

MCQ

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
$1, 1$
✓
$2, 0$
D
$2, 1$

✓

Answer

Correct option: C.

$2, 0$

c
(c) 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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