What is the value of K, if the function f(x) = \(\left\{\begin{array}{cl} \rm K x^2, & \rm x \geq 1 \\ 4, & \rm x<1\end{array}\right.\) is continuous at x = 1?

Concept Used:

For a function f(x) to be continuous at x = a, the following condition needs to be met

\( \lim_{x\to a^+}f(x) =\lim_{x\to a^-}f(x) = f(a)\)

Calculation:

for f(x) = \(\left\{\begin{array}{cl} \rm K x^2, & \rm x \geq 1 \\ 4, & \rm x<1\end{array}\right.\)to be continuous at x = 1,

\(\lim_{x\to 1^+}f(x) =\lim_{x\to 1^-}f(x) = f(1)\)

⇒ \(\lim_{x\to 1^+}(Kx^2) =\lim_{x\to 1^-}(4) = K\times(1)^2\)

Taking limit, we get

\(K=4= K\)

Hence, K = 4.