Question

Let the equation $e^{-x}+2=k,k\in Z$ has atleast one real solution in $R$, then minimum value of $k$ is

Answer 1

Let $f\left(x\right)=e^{-x}+2\in (2,\infty )$ and continuous in $R$.
According to $I.V.T.$
There exists atleast one real solution of $f\left(x\right)=k,k\in (2,\infty )$
$\therefore k_{min}=3$