Question

Find the values of a and b such that the function defined by
f(x)=\left\\{\begin{matrix} 5, &if\,x\leq2 \\\ ax+b,&if\,2<x<10 \\\ 21,&if\,x\geq10 \end{matrix}\right.

is a continuous function.

Answer 1

f(x)=\left\\{\begin{matrix} 5, &if\,x\leq2 \\\ ax+b,&if\,2<x<10 \\\ 21,&if\,x\geq10 \end{matrix}\right.

It is evident that the given function f is defined at all points of the real line.
If f is a continuous function, then f is continuous at all real numbers.
In particular,f is continuous at x=2 and x=10 Since f is continuous at x=2, we obtain
$\lim_{x\rightarrow2^-}$ f(x)=$\lim_{x\rightarrow2^+}$f(x)=f(2)
\Rightarrow$$\lim_{x\rightarrow2^-}(5)=$\lim_{x\rightarrow2^+}$(ax+b)=5
$\Rightarrow$5=2a+b=5
$\Rightarrow$2a+b=5 ...(1)
Since f is continuous at x=10, we obtain
$\lim_{x\rightarrow10^-}$ f(x)=$\lim_{x\rightarrow10^+}$f(x)=f(10)
\Rightarrow$$\lim_{x\rightarrow10^-}(ax+b)=$\lim_{x\rightarrow10^+}$(21)=21
$\Rightarrow$10a+b=21
$\Rightarrow$10a+b=21 ....(2)
On subtracting equation (1) from equation (2),
we obtain 8a=16
$\Rightarrow$a=2
By putting a=2 in equation (1),
we obtain 2×2+b=5
$\Rightarrow$4+b=5
$\Rightarrow$ b=1

Therefore, the values of a and b for which f is a continuous function are 2 and 1 respectively.