If the function $f(x)=\begin{cases} 1+\sin \left(\frac{\pi x}{2}\right), & \text{for } -\infty < x \leq 1 \\…
If the function $f(x)=\begin{cases} 1+\sin \left(\frac{\pi x}{2}\right), & \text{for } -\infty < x \leq 1 \\ a x+b, & for 1 < x < 3 \\ 6 \tan \left(\frac{x \pi}{12}\right), & \text{for } 3 \leq x < 6 \end{cases}$ is continuous in the interval $(-\infty, 6)$, then the values of $a$ and $b$ are respectively
$f(x)=\begin{cases} 1+\sin \left(\frac{\pi x}{2}\right), & \text{for } -\infty < x \leq 1 \\ a x+b, & \text{for } 1 < x < 3 \\ 6 \tan \left(\frac{x \pi}{12}\right), & \text{for } 3 \leq x < 6 \end{cases}$
The possible values for the ordered pair $(a,b)$ is
0,2
1,1
2,0
2,1
Solution
n 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
$\(\begin{array}
\lim _{x \rightarrow 1-} f(x)=\lim _{x \rightarrow 1+} f(x) \Rightarrow 1+\sin \frac{\pi}{2}=a+b \\
\Rightarrow a+b=2
\end{array}\)$
If at \(x=3\), function is continuous, then
\(\begin{array}
\lim _{x \rightarrow 3^{-}} f(3)=\lim _{x \rightarrow 3^{+}} f(x) \Rightarrow 3 a+b=6 \tan \frac{3 \pi}{12} \\
\Rightarrow 3 a+b=6
\end{array}\)
From Eqs. (1) and (2), a = 2, b = 0