Let's first draw the graph of the function. Then we will use the graph to identify the increasing and decreasing intervals.

Graphing the Function

Consider the given function.

y = 0.5 x^{3} + x^{2} - x + 2

Let's draw the graph. If you need a step-by-step explanation on graphing a function using a table of values, please refer to the extra information at the bottom of this exercise.

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

	$x -$ intercepts	Relative minimum and maximum
Definition	$x -$ coordinates of the points at which the graph intersects the $x -$ axis	The lowest and highest point for a region of the graph.
In the Given Function	$- 3.07$	$(- 1.72, 4.13)$ and $(0.39, 1.79)$

Identifying 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.

As seen on the graph, the function has one local maximum and one local minimum.

Local Maximum : Local Minimum : (- 1.72, 4.13) (0.39, 1.79)

Determining 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 $x -$ direction.

This graph decreases for values of

x

greater than or equal to

- 1.72

and less than or equal to

0.39

. It increases for values of

x

less than

- 1.72

and greater than

0.39

. Notice the graph is never constant.

Decreasing Interval : Increasing Interval : - 1.72 \leq x \leq 0.39 x \leq - 1.72 and x \geq 0.39

Showing Our Work

Graphing the function using a table of values

We will plug in $x -$ values to get the $y -$ values.

$x$	$0.5 x^{3} + x^{2} - x + 2$	$y = 0.5 x^{3} + x^{2} - x + 2$
$- 3$	$0.5 (- 3)^{3} + (- 3)^{2} - (- 3) + 2$	$0.5$
$- 2$	$0.5 (- 2)^{3} + (- 2)^{2} - (- 2) + 2$	$4$
$- 1$	$0.5 (- 1)^{3} + (- 1)^{2} - (- 1) + 2$	$3.5$
$0$	$0.5 (0)^{3} + (0)^{2} - (0) + 2$	$2$
$1$	$0.5 (1)^{3} + (1)^{2} - (1) + 2$	$2.5$

We can plot these points, and connect them to get the graph of the function.

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