The sum of values of a and b such that the function $f(x)$ defined by $f(x)=\left\{\begin{matrix}3,&x≤1\\ax+b,&1<x<5\\10,&x≥5\end{matrix}\right.$ is a continuous function is
Answer & explanation
Correct answer: option 3
$f(x)=\left\{\begin{matrix}3,&x≤1\\ax+b,&1<x<5\\10,&x≥5\end{matrix}\right.$
$f(1)=3$
$\underset{x→1}{\lim}=a+b$
$⇒a+b=3$ ...(1)
$f(5)=10$
$\underset{x→5}{\lim}=5a+b$
$⇒5a+b=10$ ...(2)
Subtracting eq. (1) from eq. (2)
$4a=7⇒a=\frac{7}{4}$ from (1) $b=3-\frac{7}{4}=\frac{5}{4}$
$a+b=3$