# The function f(x) is defined as follows:

Question:

The function f(x) is defined as follows:

$f(x)= \begin{cases}x^{2}+a x+b, & 0 \leq x<2 \\ 3 x+2 & , \quad 2 \leq x \leq 4 \\ 2 a x+5 b \quad & 4 If$f$is continuous on$[0,8]$, find the values of$a$and$b$. Solution: Given:$f$is continuous on$[0,8]$.$\therefore f$is continuous at$x=2$and$x=4$At$x=2$, we have$\lim _{x \rightarrow 2^{-}} f(x)=\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0}\left[(2-h)^{2}+a(2-h)+b\right]=4+2 a+b\lim _{x \rightarrow 2^{+}} f(x)=\lim _{h \rightarrow 0} f(2+h)=\lim _{h \rightarrow 0}[3(2+h)+2]=8$Also, At$x=4$, we have$\lim _{x \rightarrow 4^{-}} f(x)=\lim _{h \rightarrow 0} f(4-h)=\lim _{h \rightarrow 0}[3(4-h)+2]=14\lim _{x \rightarrow 4^{+}} f(x)=\lim _{h \rightarrow 0} f(4+h)=\lim _{h \rightarrow 0}[2 a(4+h)+5 b]=8 a+5 bf$is continuous at$x=2$and$x=4\therefore \lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)$and$\lim _{x \rightarrow 4^{-}} f(x)=\lim _{x \rightarrow 4^{+}} f(x)\Rightarrow 4+2 a+b=8$and$8 a+5 b=14\Rightarrow 2 a+b=4 \quad \ldots(1)$and$8 a+5 b=14 \quad \ldots(2)$On simplifying eqs. (1) and (2), we get$a=3$and$b=-2\$