Consider the following.

f$($x$)=163(1$ x$2)4/3$

$($a$)$ Find f$\text{'}($x$).$

f$\text{'}($x$)=$

$($b$)$ Graph both f$($x$)$ and f$\text{'}($x$)$ with a graphing utility.

The x y$-$coordinate plane is given. There are $2$ curves on the graph.

The first curve enters the window at the approximate point $(2\mathrm{.}9,27\mathrm{.}3),$ goes down and right, crosses the x$-$axis at approximately x $=2\mathrm{.}1,$ changes direction at the point $(1,16),$ goes up and right, changes direction at the point $(0,13),$ goes down and right, changes direction at the point $(1,16),$ goes up and right, crosses the x$-$axis at approximately x $=2\mathrm{.}1,$ and exits the window at the approximate point $(2\mathrm{.}9,27\mathrm{.}3).$

The second curve enters the window at the approximate point $(2\mathrm{.}3,29\mathrm{.}9),$ goes up and right, crosses the x$-$axis almost vertically at x $=1,$ goes up and right, changes direction at the approximate point $(0\mathrm{.}8,4\mathrm{.}6),$ goes down and right, crosses the y$-$axis at the origin, changes direction at the approximate point $(0\mathrm{.}8,4\mathrm{.}6),$ goes up and right, crosses the x$-$axis almost vertically at x $=1,$ goes up and right, and exits the window at the approximate point $(2\mathrm{.}3,29\mathrm{.}9).$

The x y$-$coordinate plane is given. There are $2$ curves on the graph.

The first curve enters the window at the approximate point $(2\mathrm{.}9,27\mathrm{.}3),$ goes up and right, crosses the x$-$axis at approximately x $=2\mathrm{.}1,$ changes direction at the point $(1,16),$ goes down and right, changes direction at the point $(0,13),$ goes up and right, changes direction at the point $(1,16),$ goes down and right, crosses the x$-$axis at approximately x $=2\mathrm{.}1,$ and exits the window at the approximate point $(2\mathrm{.}9,27\mathrm{.}3).$

The second curve enters the window at the approximate point $(2\mathrm{.}3,29\mathrm{.}9),$ goes up and right, crosses the x$-$axis almost vertically at x $=1,$ goes up and right, changes direction at the approximate point $(0\mathrm{.}8,4\mathrm{.}6),$ goes down and right, crosses the y$-$axis at the origin, changes direction at the approximate point $(0\mathrm{.}8,4\mathrm{.}6),$ goes up and right, crosses the x$-$axis almost vertically at x $=1,$ goes up and right, and exits the window at the approximate point $(2\mathrm{.}3,29\mathrm{.}9).$

The x y$-$coordinate plane is given. There are $2$ curves on the graph.

The first curve enters the window at the approximate point $(2\mathrm{.}9,27\mathrm{.}3),$ goes down and right, crosses the x$-$axis at approximately x $=2\mathrm{.}1,$ changes direction at the point $(1,16),$ goes up and right, changes direction at the point $(0,13),$ goes down and right, changes direction at the point $(1,16),$ goes up and right, crosses the x$-$axis at approximately x $=2\mathrm{.}1,$ and exits the window at the approximate point $(2\mathrm{.}9,27\mathrm{.}3).$

The second curve enters the window at the approximate point $(2\mathrm{.}3,29\mathrm{.}9),$ goes down and right, crosses the x$-$axis almost vertically at x $=1,$ goes down and right, changes direction at the approximate point $(0\mathrm{.}8,4\mathrm{.}6),$ goes up and right, crosses the y$-$axis at the origin, changes direction at the approximate point $(0\mathrm{.}8,4\mathrm{.}6),$ goes down and right, crosses the x$-$axis almost vertically at x $=1,$ goes down and right, and exits the window at the approximate point $(2\mathrm{.}3,29\mathrm{.}9).$

The x y$-$coordinate plane is given. There are $2$ curves on the graph.

The first curve enters the window at the approximate point $(2\mathrm{.}9,27\mathrm{.}3),$ goes up and right, crosses the x$-$axis at approximately x $=2\mathrm{.}1,$ changes direction at the point $(1,16),$ goes down and right, changes direction at the point $(0,13),$ goes up and right, changes direction at the point $(1,16),$ goes down and right, crosses the x$-$axis at approximately x $=2\mathrm{.}1,$ and exits the window at the approximate point $(2\mathrm{.}9,27\mathrm{.}3).$

The second curve enters the window at the approximate point $(2\mathrm{.}3,29\mathrm{.}9),$ goes down and right, crosses the x$-$axis almost vertically at x $=1,$ goes down and right, changes direction at the approximate point $(0\mathrm{.}8,4\mathrm{.}6),$ goes up and right, crosses the y$-$axis at the origin, changes direction at the approximate point $(0\mathrm{.}8,4\mathrm{.}6),$ goes down and right, crosses the x$-$axis almost vertically at x $=1,$ goes down and right, and exits the window at the approximate point $(2\mathrm{.}3,29\mathrm{.}9).$

$($c$)$ Determine x$-$values where

f$\text{'}($x$)=0.$

$($Enter your answers as a comma$-$separated list.$)$

x $=$

Determine x$-$values where

f$\text{'}($x$)>0,$ f$\text{'}($x$)<0.$

$($Enter your answers using interval notation.$)$

f$\text{'}($x$)>0$

f$\text{'}($x$)<0$

$($d$)$ Determine x$-$values for which f$($x$)$ has a maximum or minimum point. $($Enter your answers as a comma$-$separated list.$)$

maximum point

x $=$

minimum point

x $=$

Determine where the graph of f$($x$)$ is increasing, and where it is decreasing. $($Enter your answers using interval notation.$)$

increasing

decreasing

$.$