Challenge: Solving the Optimisation Problem

Let's consider a physics-related optimization problem where we need to find the maximum height reached by an object thrown vertically upward with a given initial velocity.

We have the following equation:
h = v * t - 0.5 * g * t**2
that describes the motion of an object.

Our task is to find the time t when the object reaches its maximum height and then find the maximum height h_max.

1. Calculate the derivatives of the first and second order for the h function.
2. Find critical points of h function.
3. Check if these critical points are points of the maximum of the function h.