I have two functions $k(t)$ and $l(t)$ in a certain closed interval $[a,b]$ both functions are continuous and differentiable in the interval. In addition we have:
- Both functions are increasing with bounded derivatives in the interval $ A_{max} > k'(t)> A_{min} >0$ and $B_{max} > l'(t)>B_{min}>0, \forall t \in [a,b]$
- The functions and derivatives verify that $\frac{l'(t)}{k'(t)}<\frac{l(t)}{k(t)} \ \forall t \in [a,b]$, which implies that $l(t)<k(t)\ \forall t$ (you can prove it by integrating the first inequality on $t$)
- There is a point of the interior, $c \in (a,b)$, where both functions are zero $k(c)=l(c)=0$ (i.e. they change simultaneously their signs)
Now with this information I can affirm that the quotient $\frac{l(t)}{k(t)}$ is continuous in $[a,b]$, because the only problematic point is $\frac{l(c)}{k(c)} = \frac{0}{0}$, but according to l'Hôpital $\lim_{t\to c}\frac{l(t)}{k(t)} = \lim_{t\to c}\frac{l'(t)}{k'(t)}$ which exists because the derivatives have bounded positive values. Ok.
But the question is:
Is the quotient $\frac{l(t)}{k(t)}$ also differentiable in $[a,b]$? or equivalently Is the quotient $\frac{l(t)}{k(t)}$ differentiable in $c \in [a,b]$? If not, Is there any counter example? What would be then the minimum conditions for differentiability in $c$ (or in $[a,b]$) to be proven?
Thanks a lot, I have been working on that for some days, but no result.