could you help me on the following economics proof ?

Imagine a policymaker with the following utility function: U(S,g, W) = f(Z(S+g) ~Z(S)) + w 9 Where 2' if a concave, increasing function representing the production function of a public good, f is a concave, increasing function representing the policymaker's utility from the impact on the public good, Sis a random variable (symmetrically distributed) which constitutes the exogenous level of resources for the public good other than what the policymaker invests, g represents the amount the policymaker contributes to the public good (before they know the value of 5'), w represents the policymaker's budget. Z (S + g) Z (5') represents the impact the policymaker has on the public good. Please prove that g is decreasing in the variance of S(given the same mean of S), or prove that g is decreasing in Sunder some circumstances/assumptions which you can specify to make the problem more tractable (e. g., you can also assume Zm = 0 and/orf' ' ' = 0, or even assume specic functional forms or simple distributions of S e.g., binomial). Now, with similar assumptions to those you used for the last problem, please prove that g is increasing in the variance of S for this modified utility function where f represents utility from the total level of the public good 2(5) + g): U(519.W) = f(Z(S+g)) + w -9