To solve this problem, let's break down each part.

Part (a): Determine Concavity

The question requires identifying intervals where the function $f$ is:

1. Concave Upward: where the slope of the tangent (first derivative) is increasing.
2. Concave Downward: where the slope of the tangent (first derivative) is decreasing.

From the graph, concave upward and downward intervals can be determined by observing changes in the curve's direction:

• Concave Upward (where the curve bends upwards): Typically found between points of inflection.
• Concave Downward (where the curve bends downwards): Also typically found between inflection points.

Part (b): Find Inflection Points

Inflection points are where the concavity of the function changes. These can be identified as points where the curve shifts from bending upward to downward (or vice versa).

Let's analyze the graph closely:

1. Concave Upward Intervals:

   • From about $x = 2$ to $x = 4$.
   • From about $x = 7$ to $x = 10$.

2. Concave Downward Intervals:

   • From about $x = 4$ to $x = 7$.
   • From about $x = 10$ to $x = 12$.

So, the intervals in interval notation are:

• Concave Upward: $(2, 4) \cup (7, 10)$
• Concave Downward: $(4, 7) \cup (10, 12)$

Inflection Points:

Inflection points are approximately where $x$ changes in concavity:

• $x \approx 4$
• $x \approx 7$
• $x \approx 10$

Using the $y$-coordinates from the graph:

• Inflection points are approximately at: $(4, 2.5)$, $(7, 3)$, and $(10, 6)$.

These points are based on the graph provided and might need slight adjustments if exact values are required.

Would you like a more detailed explanation of concavity or inflection points, or do you have further questions?

Here are some related questions you might find helpful:

1. What is the significance of concavity in calculus?
2. How do we find the concavity of a function using its second derivative?
3. What are the conditions for a point to be an inflection point?
4. How does concavity relate to the behavior of the first derivative?
5. Can a function have multiple inflection points in a single interval?

Tip: Remember, concave upward and concave downward intervals can be determined visually by the shape of the curve, but precise values often require calculation with derivatives.