I have two functions $T\_1(x)$ and $T\_2(x)$ defined on the interval $[0,1]$ such that $T\_1(0) = T\_2(0) = 0$ and $T\_1(1) = T\_2(1) = 1$. Furthermore, both of these are strictly monotone functions and are convex in the given interval. Also, they are differentiable at least twice. My aim is to prove these two functions are exactly equal. I have the following additional constraints on them.

1. $T\_1'(0) = T\_2'(0)$ and $T\_1'(1) = T\_2'(1)$
2. They intersect at $2$ points in (0,1).

Intuitively, I think that the functions should be the same given the constraints and I think only the first condition is sufficient to prove this. However, I am not able to get a proof for the same even with both these conditions. I tried a variety of things but they did not lead me anywhere. Any help on this would be highly appreciated. Thanks!