Question

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

(1) \( \frac{2}{\pi-5} \) 

(2) \( \frac{-2}{\pi+5} \) 

(3) \( \frac{2}{\pi+5} \) 

(4) \( \frac{2}{5-\pi} \)

Answer 1

Correct option is (4) \(\frac{2}{5 - \pi}\)

Function f(x) is continuous at x = 5.

\(\therefore f(5^-) = f(5^+)\)

⇒ \(a|\pi - 5|+ 1 = b|5 - \pi| + 3\)

⇒ \(- a(\pi - 5) + 1 = b(5 - \pi) + 3\)

⇒ \(\pi (a - b) - 5(a - b) = 1 -3\)

⇒ \(a - b = \frac{-2}{ \pi- 5} = \frac2 {5 - \pi}\)