Question
Let $\lbrack t\rbrack$ denote the greatest integer less than or equal to $t$. If the function
$$f(x) = \left\{ \begin{matrix} b^{2}sin\left( \frac{\pi}{2}\left\lbrack \frac{\pi}{2}(cosx + sinx)cosx \right\rbrack \right) & \ ,x < 0 \\ \frac{sinx - \frac{1}{2}sin2x}{x^{3}} & \ ,x > 0 \\ a & \ ,x = 0 \end{matrix} \right.\ $$is continuous at $x = 0$, then $a^{2} + b^{2}$ is equal to
Options
Solution
$f(0) = a$
$$\begin{matrix} & RHL = \lim_{x \rightarrow 0^{+}}\mspace{2mu}\frac{sinx(1 - cosx)}{x^{3}} = \frac{1}{2} \\ & LHL = \lim_{x \rightarrow 0^{-}}\mspace{2mu}\left( b^{2}sin\frac{\pi}{2}\left\lbrack \frac{\pi}{2}(sinx + cosx)cosx \right\rbrack \right) = b^{2} \\ & \ \therefore a = \frac{1}{2}\&{\text{ }b}^{2} = \frac{1}{2} \\ & \text{~}\text{so}\text{~}\left( a^{2} + b^{2} \right) = \frac{1}{4} + \frac{1}{2} = \frac{3}{4} \end{matrix}$$
Create a free account to view solution
View Solution Free