Let be $t \in \mathbb{R}$, $n = 1,2,\cdots$, $p \in [0,1]$ and $a\in (p,1]$.
Show that $$\sup_{t}\left(ta - \log (pe^t + (1-p)\right) = a\log\left(\frac{a}{p}\right) + (1-a)\log\left(\frac{1-a}{1-p}\right)$$
So, I've try to derivate, but I did'nt get sucess, since my result is different. I've get $\log\left(\frac{a(1-p)}{p(1-a)}\right)$. Any ideas?
You are given a function $f$ of the variable $t$, with two parameters $p$ and $a$. You are asked to verify that the supremum of the function has a certain specific form in terms of the two parameters.
The correct procedure to find the supremum (or maximum) is to differentiate the function $f(t)$ with respect to the variable $t$. This give you the first derivative. The next step is to set the first derivative equal to zero, and solve the equation in terms of $t$. From your post I see that you have done so, and found that $t = log(a) - log(1-a) + log(1-p) -log(p)$. This answer is correct !
Now all you have to do is substitute this particular value fot $t$ into the expression for $f(t)$. This step will yield the supremum. If you perform this calculation (try it!), you will indeed find the result that was specified in the exercise.