I figured it out.
F''(t) can be written as X'HX where X=(X1_i -X_2_i) For i = 1,2.....n
and H is the hessian
Therefore F''(t) >= 0 is equivalent to H being positive definite
which in turn implies that F(t)<= 0 for 0<=t<=1... This gives us convexity of f(.)