At what time is the speed minimum? The position function of a particle is given by $r(t) = \langle-5t^2, -1t, t^2 + 1t\rangle$. At what time is the speed minimum?

The velocity vector is $\left<-10t,-1,2t+1\right>$. Thus the speed is $\sqrt{(-10t)^2 +(-1)^2+(2t+1)^2}$.
We want to minimize this, or equivalently we want to minimize $104t^2+4t+2$. This is a problem that can even be solved without calculus, by completing the square. 
If negative $t$ is allowed, we get $t=-\frac{4}{208}$. If by nature $t\ge 0$, the minimum speed is at $t=0$.
