What is the minimum time for light to travel through two mediums?

[Annes]

i simply can't figure out how to make the equations work, so i copied everything into pictures.

 

[BruceW]

dt/dx=0 will give local stationary points (minima or maxima). In the general case, there may be several points which satisfy dt/dx=0. So to actually prove that a point is the lowest possible t value, you would need to draw a graph and/or use good reasoning.

I don't understand why you were taking the limit of dt/dx as x goes to zero. This would give the case where the light goes the shortest distance through the first medium. And taking the limit as x goes to infinity makes no physical sense, because we would expect x<m.

In this problem, most teachers expect that you just try to solve dt/dx, rather than go into detail about why this minimises t. I don't know what your teacher is looking for, but I would guess he just wants you to solve dt/dx and provide a reasonable explanation to why it minimises t.