Create a graph of a function given the following information:

The instantaneous rate of change at x = 2 is zero.
The instantaneous rate of change at x = 3 is negative.
The average rate of change on the interval 0 <x <4 is zero.

The instantaneous rate of change at x = 2 is zero.

Means: A peak trough or inflection. The curve levels off there; the tangent is paralel to the x-axis.

The instantaneous rate of change at x = 3 is negative.
Means: There's a downward slop at x=3

The average rate of change on the interval 0 <x <4 is zero.
Means: It's symmetric over the interval - for every up there is a down, and you end as high as you start.

Thus: It's a hill. Plot a parabola with a peak at x=2.

Eg. Plot a curve between points:
(0,0) (1,3) (2,4) (3,3) (4,0)

Good answer. They didn't actually ask for the function, so the above points (and also many other sets) will work.

To create a graph of the function based on the given information, follow these steps:

Step 1: Find the y-value at x = 2 and x = 3 using the given information.
Since the instantaneous rate of change is zero at x = 2, this indicates that the function has a local maximum or minimum at that point. We'll use y = 0 as the y-value at x = 2.

Since the instantaneous rate of change at x = 3 is negative, this implies that the function is decreasing at that point. We'll use a negative y-value, such as y = -1, at x = 3.

Step 2: Find two additional points on the graph using the given average rate of change information.
The average rate of change on the interval 0 < x < 4 is zero, which means the graph is flat over that interval. We'll use the points (1, 0) and (4, 0) to represent the flat part of the graph.

Step 3: Plot the identified points on the graph.
Now, plot the points (2, 0), (3, -1), (1, 0), and (4, 0) on the xy-plane.

Step 4: Connect the points to create the graph.
Once all the points are plotted, connect them smoothly to create the graph. The portion between x = 0 and x = 4 should be flat, representing the average rate of change on that interval. The graph will have a local maximum or minimum at x = 2 and will be decreasing at x = 3.

The resulting graph will have a flat section between x = 0 and x = 4 with the points (1, 0) and (4, 0). The graph will also have a local maximum or minimum at x = 2 with the point (2, 0) and will be decreasing at x = 3 with the point (3, -1).

To create a graph of a function based on the given information, we can follow these steps:

Step 1: Determine the behavior of the function at x = 2:
Since the instantaneous rate of change at x = 2 is zero, this indicates that the function has either a maximum or minimum point at x = 2. To visually represent this on the graph, we can plot a point at (2, f(2)).

Step 2: Determine the behavior of the function at x = 3:
Since the instantaneous rate of change at x = 3 is negative, this suggests that the function is decreasing at x = 3. To reflect this on the graph, we can plot a point at (3, f(3)) below the point (2, f(2)).

Step 3: Determine the behavior of the function on the interval 0 < x < 4:
Given that the average rate of change on this interval is zero, it suggests that the function is symmetric around the line x = 2. This means that the shape of the graph will be the same on either side of x = 2. Therefore, we can draw a symmetrical graph using the points (2, f(2)) and (3, f(3)).

Step 4: Connect the points:
Based on the information provided, we can connect the plotted points (2, f(2)) and (3, f(3)) with a smooth curve, keeping in mind the shape and symmetry of the graph.

It's important to note that additional information, such as the domain and range of the function, is not given. As a result, we cannot determine the exact shape of the graph or the values of f(2) or f(3) without more details. The provided instructions allow us to sketch a plausible graph based on the given information.