Mr$.$ Greenthumb wishes to mark out a rectangular flower bed, using a wall of his house as one side of the rectangle. The other three sides are to be marked by wire netting, of which he has only $64ft$ available. What are the length $L$ and width $W$ of the rectangle that would give him the largest possible planting area? How do you make sure that your answer gives the largest, not the smallest area?

A firm has the following total$-$cost and demand functions:

$C=\frac{1}{3}{Q}^3-7Q^2+111Q+50$

$Q=100-P$

$($d$)$ Does the total$-$cost function satisfy the coefficient restrictions of $(9\mathrm{.}5)?$

$($b$)$ Write out the total$-$revenue function $R$ in terms of $Q.$

$($c$)$ Formulate the total$-$profit function $$ in terms of $Q.$

$($d$)$ Find the profit$-$maximizing level of output $Q^{**}.$

$($e$)$ What is the maximum profit?

$4.$ If coefficient $b$ in $(9\mathrm{.}3)$ were to take a zero value, what would happen to the marginalcost and total$-$cost curves?

$5.$ A quadratic profit function $(Q)=h{Q}^2+jQ+k$ is to be used to reflect the following assumptions:

$($a$)$ If nothing is produced, the profit will be negative $($because of fixed costs$).$

$($b$)$ The profit function is strictly concave.

$($c$)$ The maximum profit occurs at a positive output level $Q^{**}.$

What parameter restrictions are called for?

$6.$ A purely competitive firm has a single variable input $L($labor$),$ with the wage rate $W_0$ per period. Its fixed inputs cost the firm a total of $F$ dollars per period. The price of the product is $P_0.$

$($a$)$ Write the production function, revenue function, cost function, and profit function of the firm.