Explanation:

To determine whether the point (0, 0) is a point of maximum, minimum, or neither for the function f(x, y) = x^2 + y^2, we need to analyze the behavior of the function in the vicinity of the point.

Definition of Maxima and Minima:

In calculus, a point (x0, y0) is considered a point of maximum if the function takes on larger values at points nearby but smaller values farther away. Similarly, a point (x0, y0) is considered a point of minimum if the function takes on smaller values at points nearby but larger values farther away.

Analysis:

Let's analyze the behavior of the function f(x, y) = x^2 + y^2 in the vicinity of the point (0, 0).

If we consider any point (x, y) close to (0, 0), the value of the function will always be positive or zero since both x^2 and y^2 are non-negative quantities. In other words, f(x, y) ≥ 0 for all (x, y).

When we approach the point (0, 0) along the x-axis (x ≠ 0), the function simplifies to f(x, 0) = x^2. This means that the function takes on positive values for x ≠ 0. Similarly, when we approach the point (0, 0) along the y-axis (y ≠ 0), the function simplifies to f(0, y) = y^2, which also takes on positive values for y ≠ 0.

Conclusion:

Since the function takes on positive values in the vicinity of the point (0, 0) and there are no points nearby where the function takes on smaller values, we can conclude that (0, 0) is a point of minimum for the function f(x, y) = x^2 + y^2.

Therefore, the correct answer is option 'D' - minima.