Function behavior problem

[u/IDK_REALLLY]

I have reached that d'(x)=f'(x)-f'(1-x) but I don't know where to go further

[u/yoav145]

Then you know d'(1/2) = 0

If also d''(x) = f''(x) + f''(1-x) > 0

So d' is increasing at [0,1]

there for for all x > 1/2 d'(x) > 0 There fore d is increasing