$7\mathrm{.}3$ The long$-$run total cost function for a firm producing skateboards is

$TC=q^3-40q^2+430q$

where $q$ is the number of skateboards per week.

a$.$ What is the general shape of this total cost function?

b$.$ Calculate the average cost function for skate boards. What shape does the graph of this function have? At what level of skateboard output does average cost reach a minimum? What is the average cost at this level of outpue?

c$.$ The marginal cost function for skateboards is given by

$MC=3q^2-80q+430$

Show that this marginal cost curve intersects average cost at its minumum value.

d$.$ Graph the average and marginal cost curves for skareboard production.