Functions Question 236
Question: If the function $ f(x)= \begin{cases} & 1+\sin \frac{\pi x}{2},,for,-\infty <x\le 1 \\ & ax+b,,for,1<x<3 \\ & 6\tan \frac{x\pi }{12},,for3\le x<6 \\ \end{cases} . $ is continuous in the interval $ (-\infty ,,6) $ , then the values of a and b are respectively
[MP PET 1998]
Options:
A) 0, 2
B) 1, 1
C) 2, 0
D) 2, 1
Show Answer
Answer:
Correct Answer: C
Solution:
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 $ \underset{x\to {1^{-}}}{\mathop{\lim }}f(x)=\underset{x\to {1^{+}}}{\mathop{\lim }}f(x) $
$ \Rightarrow ,1+\sin \frac{\pi }{2}=a+b $
$ \therefore ,a+b=2 $ …..(i) If at $ x=3, $ function is continuous, then $ \underset{x\to {3^{-}}}{\mathop{\lim }}f(3)=\underset{x\to {3^{+}}}{\mathop{\lim }}f(x) $
$ \Rightarrow 3a+b=6\tan \frac{3\pi }{12} $
$ \therefore ,3a+b=6 $ …..(ii) From (i) and (ii), $ a=2,b=0 $ .
BETA
