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Suppose we have random variable $X$ and functions $f(X)$ and $g(X)$. I need to know that if $f(X)$ and $g(X)$ are both increasing then the solution of the following maximization has two mass points.?! $$\max_{P(X),\\ \text{Subject to}\quad 0\leq X \leq A, \quad E[X]\leq\alpha}Cov(f(X),g(X))$$ Remark: It could be shown if $f(x)$ or $g(X)$ are a linear function of $X$, then the solution of maximization of covariance by considering peak and average constraints has two mass points. For example, if $f(X)=X$ and $g(X)$ is increasing, then the solution has two mass points at $X=0$ and $X=A$.

Thank you for your help

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    So you optimize over the probability density/mass of $X$? I do not see why $X=0$ is feasible given that $0$\alpha$ does not affect the solution. – 2017-01-01
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    @LinAlg Thank you for your comment. I have corrected the typos. – 2017-01-04
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    Still the solution does not depend on $\alpha$? – 2017-01-04
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    If the OPs claim is true then I guess the probabilites would. E.g., for ${\rm Pr}[X=A] = \frac{\alpha}{A}$ we would get $E[X]=a$. – 2017-01-04

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