Problem $9$

If $f(x)=x^3-3x^2,$ find the solutions to $f^{\text{'}}(x)=0.$

Problem $10$

Compute the first$-$ and second$-$order derivatives for the following functions:

$($a$)y=2x-5$

$($b$)y=\frac{1}{3}{x}^9$

$($e$)y=\frac{1}{10}(x-5)$

$($f$)y=x^5-x^{-5}$

Problem $11$

Answer the questions:

$($a$)$ Let $C(Q)$ denote the cost of producing $Q$ units per month of a commodity. What is the interpretation of $C^{\text{'}}(1000)=25?$ Suppose the price obtained per unit is fixed at $30$ and that the current output per month is $1000.$ Why is it profitable $($or not profitable$)$ to increase production?

You explanation:

$($b$)$ A firm's profit function is $(Q)=24Q-Q^2-5.$ Find the marginal profit and the value $Q^{**}$ of $Q,$ which maximizes profits.

$($c$)$ A firm's revenue function is $R(Q)=500Q-\frac{1}{3}{Q}^3.$ Find the marginal revenue.

$($d$)$ For the particular cost function $C(Q)=-Q^3+214\mathrm{.}2Q^2-7900Q+320700,$ find the marginal cost.

$4$ of $6$

Problem $12$

Differentiate the following functions w$.$r$.$t$.x$ :

$($a$)x+1$

$($b$)x+x^2$

$($c$)3x^5+2x^4+5$

$($d$)8x^4+2\sqrt[\phantom{2}]{x}$

$($e$)\frac{1}{2}x-\frac{3}{2}{x}^2+5x^3$

$($f$)1-3x^7$