Question

Redo Problem 89 with the following additional information: increasing on $(-\infty ,-1),(2,\infty )$; decreasing on $(-1,2)$. Again compare your graph with others and comment on any

differences.

Short answer

The required graph is shown below:

The graph is increasing on $(-\infty ,-1),(2,\infty )$and decreasing on $(-1,2)$satisfies the given additional properties.

Step-by-step solution

Step 1. Given information

The following properties are given.

• Domain: All real numbers; range: all real numbers;
• Intercepts: $(0,-3)$and $(3,0)$;
• A local maximum value of $-2$is at $-1$; a local minimum value of localid="1647348785165" $-6$is at localid="1647348744096" $2$.

The additional following properties are given.

increasing on $(-\infty ,-1),(2,\infty )$; decreasing on $(-1,2)$.

Step 2. Draw the figure with the given properties.

First, plot the points $(0,-3),(3,0),(-1,-2),$and $(2,-6)$on a cartesian plane. Then, connect the points to make the graph.

It is given that the domain and range are all real numbers.

This implies that there is no asymptotes.

Since there is one local maxima and one local minima, the graph will be the graph of the cubic polynomial.

Thus, the required graph is shown below:

Step 3. Determine whether the graph satisifies the given additional properties. 

From the graph, observe that the graph is increasing on $(-\infty ,-1),(2,\infty )$and decreasing on $(-1,2)$.

Thus, conclude that the graph satisfies the given additional properties.