Correct Answer - Option 1 : 3 / 4
Concept:
A function f(x) is said to be continuous at a point x = a, in its domain if exists or its graph is a single unbroken curve.
f(x) is Continuous at x = a ⇔ \(\rm \lim_{x\rightarrow a^{+}}f(x)=\lim_{x\rightarrow a^{-}}f(x)=\lim_{x\rightarrow a}f(x)\)
Calculation:
\(\rm f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {Kx^2}&{if}&{x \le 2}\\ 3&{if}&{x > 2} \end{array}} \right.\)
\(\rm \lim_{x\rightarrow 2^{-}}f(x)=\lim_{x\rightarrow 2}K x^{2}\)
⇒ \(\rm \lim_{x\rightarrow 2^{-}}f(x)=4 K\)
Similarly,
\(\rm \lim_{x\rightarrow 2^{+}}f(x)=3 \)
Function is continuos at x = 2,
So, \(\rm \lim_{x\rightarrow 2^{+}}f(x)=\lim_{x\rightarrow 2^{-}}f(x)=\lim_{x\rightarrow 2}f(x)\)
⇒ 4K = 3
⇒ K = 3/4
The correct option is 1.