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Question
Write the value of b for which \[f\left( x \right) = \begin{cases}5x - 4 & 0 < x \leq 1 \\ 4 x^2 + 3bx & 1 < x < 2\end{cases}\] is continuous at x = 1.
Solution
Given,
\[f\left( x \right) = \begin{cases}5x - 4 & 0 < x \leq 1 \\ 4 x^2 + 3bx & 1 < x < 2\end{cases}\]
If \[f\left( x \right)\] is continuous at \[x = 1\] , then
\[\lim_{x \to 1^-} f\left( x \right) = \lim_{x \to 1^+} f\left( x \right) = f\left( 1 \right) \left[ \text{ From eq } . (1) \right]\]
\[ \Rightarrow 1 = 4 + 3b = 1\]\[\Rightarrow 1 = 4 + 3b\]
\[ \Rightarrow - 3 = 3b\]
\[ \Rightarrow b = - 1\]
Thus, for
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