velocity at the peak point of the curve

Question

Why is it so that in the velocity time graph, even though the derivative of the function at a point x is zero, the velocity is said to be maximum:

Answer 1

if we take the function $v(t)=s'(t)$ describing the velocity, then $v'(t)=s''(t)$ describes the acceleration and $v'(t)=0$ means (of course if $\exists \delta>0 \forall \epsilon<\delta\,: v'(t-\epsilon)\cdot v'(t+\epsilon)\leq 0$), that the velocity reaches the local extremum (maximum or minimum).

On the other hand, If we have a function $s(t)$ describing dependence of distance and time, then $s'(t)$ describes the velocity. $s'(t) =0$ means then, that the velocity is equal to $0$, not maximum. This is for the distance time graph.