Exercise 28 Page 217

We want to find the $x\text{-}$intercepts, and the local minimum and maximum. Let's think about what these terms mean and then obtain the desired information.

Identify $x$-intercepts and Local Extrema

A local maximum is the point where the function goes from increasing interval to decreasing interval. A local minimum is the point where the function goes from decreasing interval to increasing interval.

Determine the Intervals

To determine the intervals for which the function is increasing, decreasing, and constant, let's consider the graph of the function as we move along it in the positive $x\text{-}$direction.

Showing Our Work

Graphing the function using a table of values

To graph the function, we will use a table to plug in $x\text{-}$values and get the corresponding $y\text{-}$values.

$x$	$0.7x^4-3x^3+5x$	$f(x)=0.7x^4-3x^3+5x$
$\text{-}1$	$0.7(\text{-}1{)}^4-3(\text{-}1{)}^3+5(\text{-}1)$	$\text{-}1.3$
$0$	$0.7(0{)}^4-3(0{)}^3+5(0)$	$0$
$1$	$0.7(1{)}^4-3(1{)}^3+5(1)$	$2.7$
$3$	$0.7(3{)}^4-3(3{)}^3+5(3)$	$\text{-}9.3$

We will now plot the obtained points in a coordinate plane and connect them with a non-linear line.