Question

If the function $f\left(x\right)$ given by $f\left(x\right)=\{\begin{array}{cc}3ax+b& ifx>1\\ 11& ifx=1\\ 5ax-2b& ifx<1\end{array}$ is continuous at x=1, then find the values of a and b.

Answer 1

Here, $f\left(x\right)=\{\begin{array}{cc}3ax+b& ifx>1\\ 11& ifx=1\\ 5ax-2b& ifx<1\end{array}$

Now RHL, $\underset{x\to 1^{+}}{lim}3ax+b$

$\underset{h\to 0}{lim}3a(1+h)+b$

$=3a+b$

Again, LHL , $\underset{x\to 1^{-}}{lim}5ax-2b$

$\underset{h\to 0^{-}}{lim}5a(1-h)-2b$

$=5a-2b$

Now, RHL = LHL $=11$

$3a+b=11$........(i) $\times 2$

$5a-2b=11$.......(ii) $\times 1$

$6a+2b=22$

$5a-2b=11$

_____________

$11a=33$

$a=3$

$\therefore 3\times 3+b=11$

$b=11-9=2$

$a=3,b=2$