In this question, the answers are determined by analyzing the local behavior of the function $f(x)$ at specific points $x = c$ and $x = d$.

Part 1: Analysis at $x = c$

The answer already chosen is B. a local maximum. This implies that $f(x)$ has a peak at $x = c$, where the function value at $x = c$ is greater than the values of $f(x)$ in its immediate neighborhood. This answer seems correct if the function’s derivative changes from positive to negative at $x = c$ or if there’s other supporting evidence of a maximum at this point.

Part 2: Analysis at $x = d$

The options for $x = d$ include:

• A. a local maximum
• B. no local extremum
• C. a local minimum

To determine the correct answer for $x = d$, we would need additional information about the behavior of $f(x)$ around $x = d$:

• If $f(x)$ has a peak or is higher than its surroundings at $x = d$, the answer would be A. a local maximum.
• If $f(x)$ has a dip or is lower than its surroundings at $x = d$, the answer would be C. a local minimum.
• If there is no change in slope or it is not a distinct peak or valley, then B. no local extremum is the answer.

Without further information, it’s challenging to select an answer for $x = d$. However, based on typical interpretations, B. no local extremum could be plausible if there’s no distinct maximum or minimum.

Would you like detailed steps on how to determine extremum points or have any questions on this?

Here are some related questions:

1. How can we determine if a function has a maximum or minimum at a point?
2. What role do derivatives play in identifying local extrema?
3. How does the second derivative test help in identifying maxima and minima?
4. What are the conditions for a point to be classified as having no local extremum?
5. How do we differentiate between absolute and local extrema?

Tip: When analyzing a function’s behavior, check for sign changes in the first derivative to determine local maximum and minimum points.