Problem $5$

Examine where the following functions are increasing$/$decreasing $($use the result in $(6\mathrm{.}3\mathrm{.}6))$:

$($a$)f(x)=x^2-6x+5$

$($b$)f(x)=-2(x+2)(x-6)$

$2$ of $6$

Problem $6$

Determine the following limits:

$($a$)\underset{x0}{lim}(x^2-e^x+3)$

$($b$)\underset{x2}{lim}\frac{2x+3}{x+2}$

$($c$)\underset{x-1}{lim}\frac{4-x^3}{x+2}$

$($d$)\underset{x0}{lim}\sqrt[\phantom{2}]{x+16}$

$($e$)\underset{x3}{lim}\frac{1-3x}{2-x}$

Problem $7$

Find $y^{\text{'}}$ when

$($a$)y=\frac{3}{x^3}$

$($b$)y=5x^{\frac{6}{5}}$

$($c$)y=\frac{6}{x^2{x}^{-\frac{1}{3}}}$

Problem $8$

We had this one in class.

A firm produces $f(L)=\sqrt[\phantom{2}]{L}+L$ units of a commodity when $L$ units of labour are employed. The price obtained per unit is $p$ and price per unit of labour is $w.$

$($a$)$ What is the profit function $(L)?$

$($b$)$ Find the value of $L$ for which ${}^{\text{'}}(L)=0.$