Guys I have the following homework problem to solve: There are 2 given points in a plane. If we take a point-like object with mass m and take it to the "higher" point what path should it go on to reach the other point in the shortest possible time. Only gravitational force affects our point-like object. My attempt to solution: Let point A has (x1, f(x1)) and point B (x2, f(x2)) coordinates. As we all know average velocity = whole distance/whole time, therefore whole time = whole distance/average velocity. So we just have to calculate the components of the equation than plug in. Average velocity: First we need velocity in the function of position which can be calculated from potential energy difference. ΔEpotential=ΔEkinetic m*g*(f(x1)-f(x))=1/2*m*v2 so v2=2*g*(f(x1)-f(x)) therefore v or more likely v(x) =√(2*g*(f(x1)-f(x))) . Now we have the velocity in the function of position, all we have to do is to calculate the average value of this function, which is vaverage=∫x1x2√(2*g*(f(x1)-f(x))) dx / (x2 - x1) Now we need the arc lenght of our f(x) curve in the intervall of x1 and x2. so swhole= ∫x1x2√(1+f2(x)) dx Now devide the whole distance with average velocity: twhole = swhole/vaverage now we would have to take the quotient of these two expressions and solve its derivate for zero. ∫x1x2√(1+f2(x)) dx * (x2 - x1)d __________________________________ = 0 ∫x1x2√(2*g*(f(x1)-f(x))) dx _______ dx But u havent managed to find the solution of this equation and im not even sure if my solution is wether correct or not. If anyone has an idea how to solve this equation or have found a problem in my solution im listening with open ears.