Consider the following family of candidate distributions on $X = \{ 1, \dots, k \}$: the distributions of form $P(x) = c \cdot \exp(t_1x+t_2x^2)$. Given a sample $x = (x_1, \dots, x_n)$, denote by $P^*$ the maximum likelihood estimate provided that it exists. Assume that each symbol of $X$ occurs in the sample $x$.
How can I show that the maximum likelihood estimate exists in this case and specify linear set $L$ of distributions on $X$ such that $P^*$ is equal to the I-projection of the uniform distribution onto $L$?