This is a question from a statistics textbook: Suppose that $X_1, \ldots, X_n$ form a random sample from the beta distribution with parameters $\alpha$ and $\beta$. Let $\theta=(\alpha,\beta)$ be the vector parameter. Show that the method of moments estimator is not the M.L.E.
Denote $\alpha$ and $\beta$ in terms of method of moments, which is $\alpha = {m_1(m_1-m_2) \over m_2-m_1^2}$ and $\beta = {(1-m_1)(m_1-m_2) \over m_2-m_1^2}$, where $m_j = {1 \over n} \sum_{i=1}^n X^j_i$ for $j \ge 1$.
How to prove that the method of moments is not the MLE?