Background
I have the following function: $H(x)=Qe^{-ax}$, where all parameters are positive.
Using Mathematica, I used the following formula to calculate $H(x)$'s curvature:
$H_{C}=\frac{|H''(x)|}{\left(1+H'(x)^2\right)^{\frac{3}{2}}}$.
I then calculated the derivative of $H_{C}$ and set the resulting expression equal to zero to acquire the following roots:
$H_{C, 1}=\frac{\ln{\left(\sqrt{2}aQ\right)}}{a}$ and $H_{C, 2}=\frac{\ln{\left(-\sqrt{2}aQ\right)}}{a}$.
Problem
I am interested in finding the maximum curvature of $H(x)$ over the interval $x \in \left[0,\infty\right]$. However, for certain parameter regimes, both roots have negative real parts.
Questions
What does it mean about the maximum curvature if the roots are negative?
And, is it possible to calculate the maximum curvature for regimes under which the roots are negative? If so, what am I doing wrong?
Any help would be greatly appreciated!