On the Instability and Critical Damping Conditions, $ktau = 1/e$ and $ktau = pi/2$ of the equation $dot{theta} = -k theta(t-tau)$

In this note, I show that it is possible to use elementary mathematics, instead of the machinery of Lambert function, Laplace Transform, or numerics, to derive the instability condition, $k tau = pi/2$, and the critical damping condition, $ktau = 1/e$, for the time-delayed equation $dot{theta} = -k theta(t-tau)$. I hope it will be useful for the new comers to this equation, and perhaps even to the experts if this is a simpler method compared to othe