1. ## maximum and minimum

The amount of current in an electrical system is given by the function
c(t)= -t^3 + t^2 + 21 t, where t is time in seconds and 0<t<5. Determine the times at which the current is maximal and minimal, and the amount of current in the system at these times.

2. Originally Posted by math123456
The amount of current in an electrical system is given by the function
c(t)= -t^3 + t^2 + 21 t, where t is time in seconds and 0<t<5. Determine the times at which the current is maximal and minimal, and the amount of current in the system at these times.
$-t^3+t^2+21t$
$\frac{dy}{dx} = -3t^2+2t+21=0$
$3t^2-2t-21=0 = (t+3)(3t+7)$
$t=3$ or $t=\frac{-7}{3}$
If you differentiate $3t^2-2t-21$ again to get the maximum and minimum you get $6t-2$. So if you put in t=3, then 6t-2 is positive, therefore it is a minimum. If you put in t =-7/3 then 6t-2 is negative, so that is a maximum.

Oh right, I didn't see that it said 0<t<5. In that case the smallest possible value you can get in that range is t=0.