Question

A body moves along a horizontal line according to s=f(t)=t^3-9t^2+24t a) when is s increasing and when it is decreasing? b) when v is increasing and when it is decreasing?

Answer 1

Question a) is nothing more than another way of asking "what is the derivative of f(t) and what are its roots"...
\[f^\prime(t) = 3t^{2} - 18t = 0\]

This gives us the simple solutions t=0 and t=6.
When plotting this function (type "3t^2 - 18t" in google) we can see it's a parabola that looks like this:


We can see it's positive before 0 and after 6, negative in the [0,6] interval.
These are the solutions to your question: f(t) increases before 0 and after 6, decreases in [0,6]...