Question $3.(10$ pts$.)$ A U$.$S$.$ firm is considering to optimize its production. It's production function is $\backslash ($ q$=5$ L$^\{0\mathrm{.}75\}$ K$^\{0\mathrm{.}25\}\backslash ),$ where q is the output, L is the number of labor, and K is the number of capital employed. The prices of labor and capital are $\backslash ($ w$=5\backslash )$ and $\backslash ($ r$=10\backslash ).$

a$)$ If the firm has a fixed amount of capital $\backslash (\backslash$mathrm$\{$K$\}=16\backslash ),$ what is its marginal product of labor? Does the firm have a diminishing marginal product of labor? Explain why or why not. $(3$ pts$.)$

b$)$ If the firm has a fixed amount of capital $\backslash (\backslash$mathrm$\{$K$\}=16\backslash ),$ what is the average product of labor? Draw both the marginal product of labor and the average product of labor on one graph. $(3$ pts $.)$

c$)$ If the firm's fixed amount of capital increases to $32$ from $16,$ how does the firm's marginal product of labor change for its $\backslash (10^\{\backslash$text $\{$th $\}\}\backslash )$ worker? How would you justify the economics behind the changing $($or constant$)$ marginal product of labor in response to the changing level of capital? $(4$ pts$.)$