Concavity problem...what am i doing wrong?

I get the concavity backwards....I get cu (- infinity,1) cd (1, infinity)

but the answer says cd (-infinity, 1) cu (1, infinity)...I might just be getting the derivitives wrong

The problem is t2/t-1

For f'(x) I get -t^2+2t/(t-1)^2 or -t(t-2)/(t-1)^2

For f''(x) I get -2(2t+1)/(t-1)^3

I set f''(x) equal to zero...i use the denominator right? (t-1)= 0...t=1

For the critical number i get 1...plug in my test points and i get it wrong..so i figure it has to be my derivatives...Can someone work out the whole problem so i can see where i'm messing up..thanks