#### Need solution for RD Sharma maths class 12 chapter Continuity exercise 8.2 question 4 subquestion (iv)

$a=7 / 2, \: \: b=-17 / 2$

Hint:

Put LHL = RHL at x = 3 , 5

Given:

$f(x)= \begin{cases}2 & x \leq 3 \\ a x+b & 3

Explanation:

At x = 3

\begin{aligned} &\lim _{x \rightarrow 3^{+}} f(x)=\lim _{x \rightarrow 3} a x+b\\ &=3 a+b\\ &f(3)=2 \qquad ....(1) \end{aligned}

\begin{aligned} &\text { R.H.L }=\lim _{x \rightarrow 3^{+}} f(x)=\lim _{h \rightarrow 0}(3+h)=\lim _{h \rightarrow 0} a(3+h)+b \\ &=3 a+b \end{aligned}

Now, f(x) is continuous at x = 3

If 3a + b =2                .....(A)

Now, at x = 5

\begin{aligned} &\lim _{x \rightarrow 5^{-}} f(x)=\lim _{x \rightarrow 5} a x+b \\ &=5 a+b \\ & f(5)=9 \end{aligned}

\begin{aligned} &\text { L.H.L } =\lim _{x \rightarrow 5} f(x)=\lim _{h \rightarrow 0}(5-h)=\lim _{h \rightarrow 0} a(5-h)+b \\ &=5 a+b \end{aligned}

Now, f(x) is continuous at x = 5

If 5a + b = 9                    .....(B)

on solving A and B we get,

$a=7 / 2, \: \: b=-17 / 2$