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.

$a=2,b=1$

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 $a=2$