If the given function $f(x)$, defined as
$f(x)=\left\{\begin{array}{cl}
5, & \text { if } x \leq 2 \\
a x+b, & \text { if } 2<x<10 \\
21, & \text { if } x ≥ 10
\end{array}\right.$
is continuous, then value of $2 a+b$ is :
Answer & explanation
Correct answer: option 3
$f(x)=\left\{\begin{array}{cc}5 & x \leq 2 \\ a x+b & 2<x<10 \\ 21 & x \geq 10\end{array}\right.$
it is continuous
for x = 2
LHL = $\lim\limits_{x \rightarrow 2^{-}} f(x)=5$
RHL = $\lim\limits_{x \rightarrow 2^{+}} f(x)=2 a+b$ so $2 a+b=5$ .......(1)
for x = 10
LHL = $\lim\limits_{x \rightarrow 10^{-}} f(x)=10a+b$
RHL = $\lim\limits_{x \rightarrow 10^{+}} f(x)=21$
$10a+b=21$ ......(2)
so eq (2) - (1)
$\Rightarrow 10 a+b =21 - (2 a+b =5)$
$8a = 16$
a = 2
putting a = 2 in eq (1)
so 2 × 2 + b = 5
b = 5 - 4
b = 1
a = 2, b = 1
2a + b = 2 × 2 + 1 = 4 + 1 = 5