3.) Solve for the system of equations which define the solution to the following un-
constrained optimization problems, and provide sufficient conditions to demonstrate the
existence of a solution. When a functional form is not specified, state the necessary (first
order) and sufficient (second order) conditions for the problem to generate the desired solu-
tion (max or min).
a.)
$\max_x f(x) = \ln(x) - c(x)$
b.)
$\max_{x,z} h(x, z) = x^\alpha z^\beta - \theta_1 x - \psi_2 z$
c.)
$\max_{x,z} \Pi(x, z) = (x^\alpha z^{1-\alpha})^{1-\gamma} - (\psi_1 x + \psi_2 z)^2$
e.)
$\max_{x,z} f(x, z) = \log(xz) - \rho(\theta_1 x + \theta_2 z)$
