What does it means when velocity is undetermined? The original problem is that a car goes into a curve that has a slope of 45° and $\mu$=1 and I need to get the maximum and minimum speed of the car to not slide up or down.
Doing my analysis I got two equations to get those velocities but... when I put $(\mu=1)$ and $(\theta=\frac{\pi}{4})$, I get the minimum velocity as a complex number and the maximum velocity is simply undetermined. The formulas I ended up with are these:
$$v_{min}=\sqrt{Rg \left( \frac{\sin\theta-\mu \cos\theta}{\cos\theta-\mu \sin\theta}\right)}$$
$$v_{max}=\sqrt{Rg \left( \frac{\sin\theta+\mu \cos\theta}{\cos\theta-\mu \sin\theta}\right)}$$
My teacher said that my analysis is correct and they gave us those values in purpose, so I just need to know what does it mean.
 A: Here's some general guidance. Whenever you solve equations to find parameters, and then the math gives no or a nonsensical answer, go back to the physics of the problem.
You get no meaningful answer for the minimum speed. So, what would be the most extreme case you can have for the minimum speed? It's zero, because that's where there's no centrifugal force to counter the sliding inwards. So when you get no answer from the equation, ask yourself "Is this even possible in the most extreme case?". To verify, check manually from more basic equations whether the car will slide at a standstill. If it does not, your answer for the minimum speed is 0. (If you find that it does slide, your math is wrong somewhere.)
For the maximum, it's a little harder to see. As a first guess (seeing that the formula gives no answer) you can try calculating the sliding manually for larger and larger speeds. If you always see the same pattern of the same terms cancelling out and there is no sliding, you can become convinced that indeed there is no maximum speed; the car will never slide, no matter how fast you go.  Of course, that's not a rigorous proof, but as a basic "does this make sense?" check, it can be enough.
