Let $[t]$ represents the greatest integer not exceeding $t$ and $\mathrm{C}=1-2 \mathrm{e}^2$. If the…
Let $[t]$ represents the greatest integer not exceeding $t$ and $\mathrm{C}=1-2 \mathrm{e}^2$. If the function
$f(x)=\left\{\begin{array}{cc}
{\left[e^x\right],} & x < 0 \\
a e^x+[x-2], & 0 \leq x < 2 \\
{\left[e^{-x}\right]-C,} & x \geq 2
\end{array}\right.$
is continuous at $\mathrm{x}=2$, then $\mathrm{f}(\mathrm{x})$ is discontinuous at
$\mathrm{x}=1$ only
$x=0$ and $x=1$
$x=0$ only
$x=0, x=1$ and $x=\frac{1}{2}$
Solution
$\because f(x)$ is continuous at $x=2$
$\begin{aligned}
& \therefore \mathrm{LHL}=f(2) \\
& \Rightarrow a e^2+[2-2]=\left[e^{-2}\right]-\mathrm{C} \\
& \Rightarrow a e^2-1=0-\left(1-2 e^2\right) \Rightarrow a=2
\end{aligned}$