Concave Upward: (0,2), (4,6), (7,12) (even though the slope changes at x = 9, the slope is still increasing from 9 to 12; notice its upward curve)
Concave Downward: (2, 4), (6, 7)
Inflection Points: (2,2), (4,2), (6,2), (7,3)
Step-by-step explanation:
Recall that concave upward is when the slope is increasing, and concave downward is when the slope is decreasing.
It seems that from your answers (for part a), you are looking for when the slope changes from positive to negative (and vice versa). These two things (the rate of change of the slope and the sign of the slope) are critically different, but are easy to confuse. In other words, you're basing your answers off of maxima and minima (high points and low points on the graph).
What you really want to base your answers on are inflection points, which are points where the slope goes from becoming increasingly more negative/positive to the other.
For example, look at x = 2 on the graph. If we draw a line tangent to the graph for every x, we will see that the line will become steeper and steeper until we reach x = 2, where it will slowly start to taper off until it becomes flat at x = 3. I encourage you to look up some videos on the topic of concavity, they will show it better then I could.
If you can't do that, you can use a ruler to visualize the tangent line as you move it across the graph. When you have to start turning the ruler in the opposite direction to keep it tangent to the graph, you found an inflection point.
Inflection points are points that are on the graph, so I encourage you to keep that in mind when you are answering problems similar to that in part b.
Here is a useful resource: https://www.mathsisfun.com/calculus/concave-up-down-convex.html
Best of luck!
