In this problem, you are asked to match the contour plots to a function $f(x, y)$ and its partial derivatives $f_x(x, y)$ and $f_y(x, y)$.

Let's break down the three plots labeled A, B, and C:

1. Plot A:

• This plot shows symmetrical contours about the origin, and it appears to describe the original function $f(x, y)$.
• The contours are shaped in a way that suggests the function has a central peak or valley (a minimum or maximum at the origin), with a circular or elliptical symmetry.
• Therefore, Plot A most likely corresponds to the function $f(x, y)$.

2. Plot B:

• Plot B shows contours that have a vertical gradient, meaning the lines are more spread horizontally and tighter vertically.
• This indicates changes in the vertical direction (with respect to $y$) are more pronounced. This is typical for a partial derivative with respect to $y$, $f_y(x, y)$.
• Hence, Plot B likely represents $f_y(x, y)$.

3. Plot C:

• Plot C shows contours that are more tightly spaced in the horizontal direction (with respect to $x$) and more spread out vertically.
• This indicates that the changes are more pronounced in the horizontal direction, which matches the behavior of the partial derivative with respect to $x$, $f_x(x, y)$.
• Thus, Plot C corresponds to $f_x(x, y)$.

Conclusion:

• $f_x(x, y)$ corresponds to Plot C.
• $f_y(x, y)$ corresponds to Plot B.
• $f(x, y)$ corresponds to Plot A.

These match the options you have chosen.

Would you like any further details or clarifications? Here are 5 questions to expand on this:

1. How do contour lines reflect partial derivatives in general?
2. What does the steepness of contour lines indicate about the gradient?
3. How can symmetry in contour plots help identify functions and their derivatives?
4. What happens to the contour plot of a function near a saddle point?
5. Why is it important to understand the relationship between function behavior and its partial derivatives?

Tip: Contour plots provide a visual insight into how a function behaves, with tighter contours indicating steeper changes, which can help you quickly estimate gradients.