Question

If the function f(x) given by $f\left(x\right)=(\begin{array}{c}4\mathrm{ax}+b,\mathrm{if}x>1\\ 11,\mathrm{if}x=1\\ 3\mathrm{ax}-2b,\mathrm{if}x<1\end{array}$ is continuous at x = 1, then the value of a + b is

• A. 3
• B. 2
• C. 1
• D. 4

Answer 1

The correct option is B 2

We have,
$\left(\mathrm{LHL}\mathrm{at}x=1\right)=\underset{x\to 1^{-}}{\lim }f\left(x\right)=\underset{x\to 1^{-}}{\lim }\left(3\mathrm{ax}-2b\right)=3a-2b\left(\mathrm{RHL}\mathrm{at}x=1\right)=\underset{x\to 1^{+}}{\lim }f\left(x\right)=\underset{x\to 1^{+}}{\lim }\left(4\mathrm{ax}+b\right)=4a+b\mathrm{As}f\left(x\right)\mathrm{is}\mathrm{continuous}\mathrm{at}x=1\underset{x\to 1^{-}}{\lim }f\left(x\right)=\underset{x\to 1^{+}}{\lim }f\left(x\right)=f\left(1\right)\Rightarrow 3a-2b=4a+b=11\Rightarrow a=3\mathrm{and}b=-1a+b=2$