Question 186911
If you want to calculate the maximum height and you have the function h(t), you must find its derivative, and in this case we get h'(t)=-24t+456. Then we solve the equation 
-24t+456=0 to find the maximum and the minimum. The solution is t=19s.

If we want to proof that it is the time when the height is maximum, we calculate the second derivative, that is, h''(t)=-24, that is negative, so we have in t=19s a maximum (if the number is positive we would have found a minimum).
So when t=19s we get the maximum, that is equal to h(19)=-12(19)^2+456*19=4332m.