Find minimum distance

Question

I came across this problem in a maths exam. I solved this by taking that a light ray passes in such a way that it takes least path. But as this was a maths exam, i was wondering if it can be solved using maths?

Let $A=(0,1)$ and $B = (1,1)$ in the plane $\mathbb{R}^{2}$. Determine the length of the shortest path from A to B consisting of the line segments AP; PQ and QB, where P varies on the x-axis between the points $(0, 0)$ and $(1, 0)$ and Q varies on the line $y = 3$ between the points $(0, 3)$ and $(1, 3)$

Answer 1

You Can try using Cauchy Schwarz inequality.

Answer 2

Consider A' and B' obtain by reflection i.e. A'=(0,-1), B'=(1,5). Then you consider any path APQB and notice that its length is equal to A'PQB'. But such a length should not be larger than the distance between A' and B' and since the points P and Q that you found give equality, they are the minimum.