Find the values of $a$ and $b$ such that the function $f$ defined by $f(x)=\begin{cases} 5& \text{if}\hspace{.2cm} x \leq 2\\ ax+b& \text{if}\hspace{.2cm} 2< x<10\\ 21,& \text{if}\hspace{.2cm} x\geq 10\\ \end{cases}$ is a continuous function.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → $a=2,b=1$
$f(x)=\begin{cases} 5,& x \leq 2\\ ax+b,& 2< x<10\\ 21,& x\geq 10\\ \end{cases}$
and, f is a continuous function.
$∴\lim\limits_{x→2^-}=\lim\limits_{x→2^+}ax+b=f(2)$
$⇒5=2a+b$ ...(1)
and,
$\lim\limits_{x→10^-}ax+b=\lim\limits_{x→10^+}(21)$
$⇒10a+b=21$ ...(2)
∴ Solving (1) and (2),
$b=1$ and $a=2$