Question

$f\left(x\right)=\{\begin{array}{ccc}a{\tan }^{-1}\frac{1}{x-4}& ,& if0\le x<4\\ \frac{\pi }{2}& ,& ifx=4\\ b{\tan }^{-1}\frac{2}{x-4}& ,& if4<x\le 8\\ {\sin }^{-1}(7-x)+\frac{a\pi }{4}& ,& if6\le x\le 8\end{array}$ determine the values of $a$ and $b$ if $f\left(x\right)$ is continuous in the interval $[0,8]$.

Answer 1

Since $f\left(x\right)$ is continuous in the whole interval $[0,8]$ then it is continuous at $x=4$ also.

Then,

$\underset{x\to 4-}{lim}f\left(x\right)$ $=$$\underset{x\to 4+}{lim}f\left(x\right)$ $=f\left(4\right)$.

or, $a.\frac{-\pi }{2}=\frac{\pi }{2}=b\frac{\pi }{2}$ [ As $\underset{x\to 4+}{lim}{\tan }^{-1}\frac{1}{x-4}=\frac{\pi }{2}$ and $\underset{x\to 4-}{lim}{\tan }^{-1}\frac{1}{x-4}=-\frac{\pi }{2}]$

or, $a=-1$ and $b=1$.