Question

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.

Answer 1

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)

Answer 2

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