A company has the cost function $C(x)=0\mathrm{.}5x^{2}10x100$ and revenue function $R(x)=-0\mathrm{.}5x^{2}30x$ where $x$ is the number of units produced and sold.

a$)$ Find the profit function $P(x).$

b$)$ Find the marginal profit function $P^{\text{'}}(x).$

c$)$ Find and interpret the marginal profit function at $x=10.$

d$)$ Find any break$-$even points

A company has the cost function $C(x)=0\mathrm{.}01x^3-0\mathrm{.}4x^{2}10x200$ where $x$ is the number of units produced.

a$)$ Find the marginal cost function.

b$)$ Find and interpret the marginal cost function at $x=20.$

c$)$ Find the average cost function.

d$)$ Find the marginal average cost function.

e$)$ Find and interpret the marginal average cost function at $x=20.$