A firm's analysts estimate that the firm can manufacture a product according to the pro$-$

duction function $($LO$1,$ LO$2,$ LO$5,$ LO$6)$

$Q=F(K,L)=K^{\frac{3}{4}}{L}^{\frac{1}{4}}$

a$.$ Calculate the average product of labor, $A{P}_L,$ when the level of capital is fixed at

$81$ units and the firm uses $16$ units of labor. How does the average product of labor

change when the firm uses $256$ units of labor?

$b.$ Find an expression for the marginal product of labor, $M{P}_L,$ when the amount of

capital is fixed at $81$ units. Then, illustrate that the marginal product of labor depends

on the amount of labor hired by calculating the marginal product of labor for $16$ and

$81$ units of labor.

c$.$ Suppose capital is fixed at $81$ units. If the firm can sell its output at a price of $$200$

per unit of output and can hire labor at $$50$ per unit of labor, how many units of labor

should the firm hire in order to maximize profits?