Let \( f(x)=\left\{\begin{array}{cl}2 a x+4, & x2\end{array}\right. \) and ' \( b \) ' for which \( f(x) \) is continuous at \( x=2 \), is

Question

Let \( f(x)=\left\{\begin{array}{cl}2 a x+4, & x<2 \\ 4, & x=2 \text { then the value of 'a' } \\ b x^{2}+2, & x>2\end{array}\right. \) and ' \( b \) ' for which \( f(x) \) is continuous at \( x=2 \), is

Answer 1

\(f(2 ^- ) = \lim\limits_{x\to 2} \;2ax + 4 = 4a + 4\)

\(f(2^+) = \lim\limits_{x \to 2} \;bx^2 + 2 = 4b + 2\)

∵ \(f(x) \) is continuous at x = 2.

∴ \(f(2^-) = f(2^+) = f(2)\)

⇒ \(4a+ 4 = 4\) & \(4b + 2 = 4\)

⇒ \(4a = 0 \) & \(4b = 2\)

⇒ \(a = 0 \) & \(b = \frac12\)