The maximum (minimum) of a function on an interval will be a calculus type extremum in the interval or occur at an end point of the interval.
What do you have for the derivative?
You will notice that t=0 is one root, so take out t to leave you with a cubic. Plot the cubic to see where ther roots if any are between 0 and 10, you may find that there are no further roots in the given range.
So you have a calculus like extremum at t=0, and there will be another maximum or minimum at the other end of the interval.
So calculate S(0) and S(10) the smaller will be the global minimum of S in the interval and the larger will be the global maximum in the interval.
Finally you might just as well have plotted S(t) for t between 0 and 10 anyway.