The object change direction when the first derivative change its sign,. So we need to find the first derivative and equate this result to zero.
The first derivative is given by
[tex]s(t)=7\frac{5}{2}t^{\frac{3}{2}}-\frac{7}{2}t^{\frac{5}{2}}[/tex]
By equating this result to zero ,we have
[tex]7\frac{5}{2}t^{\frac{3}{2}}-\frac{7}{2}t^{\frac{5}{2}}=0[/tex]
This implies that
[tex]\frac{7}{2}t^{\frac{3}{2}}(5-t^)=0[/tex]
so, the first derivative change its sign when
[tex]\begin{gathered} t=0 \\ and \\ t=5 \end{gathered}[/tex]
as we can note in the following graph:
because the graphs falls when t= 5seconds.
This means that the motion objects change direction at t=5 seconds.
