Question

If the function $f\left(x\right)=\{\begin{array}{cc}a|\pi -x|+1,& x\le 5\\ & \\ b|x-\pi |+3,& x>5\end{array}$
is continuous at $x=5$, then the value of $a-b$ is

[1 mark]

• A. $\frac{2}{\pi -5}$
• B. $\frac{2}{\pi +5}$
• C. $\frac{2}{5-\pi }$
• D. $\frac{-2}{\pi +5}$

Answer 1

The correct option is C $\frac{2}{5-\pi }$

$f\left(x\right)=\{\begin{array}{cc}a|\pi -x|+1,& x\le 5\\ & \\ b|x-\pi |+3,& x>5\end{array}$
is continuous at $x=5$
$\therefore \underset{x\to 5^{-}}{lim}f\left(x\right)=\underset{x\to 5^{+}}{lim}f\left(x\right)=f\left(5\right)$
$\Rightarrow a|\pi -5|+1=b|5-\pi |+3$
$\Rightarrow a(5-\pi )+1=b(5-\pi )+3$
$\Rightarrow (a-b)(5-\pi )=2$
$\therefore a-b=\frac{2}{5-\pi }$