In this image, you are tasked with matching each function (on the left) with its derivative (on the right). Here's how we can approach this:

Key Observations:

• First Function (Top Left): The function is increasing and concave up, meaning its derivative should be positive and increasing. This suggests a derivative that is also positive and increasing (like a linear function with positive slope).

• Second Function (Second from Top): The function decreases, reaches a minimum, and then increases. At the minimum point, the derivative should be zero. This implies that the derivative will change from negative to positive, indicating a derivative with a root and changing sign.

• Third Function (Third from Top): This function has an inflection point where it changes from concave down to concave up. Its derivative should have a zero crossing (because at the inflection point, the slope is zero) and should change its behavior.

• Fourth Function (Bottom Left): This is a decreasing function that approaches zero with a concave down behavior. Its derivative should be negative and decreasing (or leveling off).

Matchings:

1. Top-left function (red, increasing) pairs with top-right graph (blue, positive and increasing).
2. Second from top-left function (red, minimum in the middle) pairs with second from top-right graph (blue, crossing zero).
3. Third from top-left function (red, inflection point) pairs with third from top-right graph (blue, crossing zero at an inflection point).
4. Bottom-left function (red, decreasing) pairs with bottom-right graph (blue, negative and concave down).

Would you like more details or any further explanation? Here are some relative questions:

1. How can you recognize a derivative graph from its parent function's concavity?
2. What does a horizontal tangent line (slope of zero) imply about the derivative?
3. How does the sign of the derivative relate to the increasing or decreasing nature of a function?
4. What role do inflection points play in determining the behavior of the derivative?
5. How does concavity affect the slope of the derivative?

Tip: Always check the critical points (where slope = 0) of a function to predict the behavior of its derivative.