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In calculus, when dealing with functions of multiple variables, like in the given function f(x, y) = x^2 - xy + y^2 + 1, partial derivatives play a crucial role. A partial derivative is a derivative taken of a function with respect to one variable while keeping the other variables constant.

For the function f(x, y), the partial derivatives – designated as f_x and f_y – represent the rate at which f changes as x or y alone changes, while the other variable is held fixed. In our example, f_x = 2x - y and f_y = -x + 2y. These derivatives are essential first steps to finding critical points where both partial derivatives equal zero, as seen in the solution.

Finding the partial derivatives correctly is key, as they serve as the foundation for further analysis in determining a function's critical points, relative extrema, and the overall behavior of the function.

The second derivative test is an invaluable tool for analyzing the nature of critical points. Once we have the critical points, like the point (2, 4) from our function, we need to determine whether each point is a local minimum, local maximum, or a saddle point. This is where the second derivative test comes in.

Based on the second partial derivatives f_{xx}, f_{yy}, and f_{xy} at the critical point, we can calculate the discriminant D = f_{xx}f_{yy} - (f_{xy})^2. Here, a positive D with f_{xx} > 0 indicates a local minimum, while a negative D would indicate a saddle point, and a positive D with f_{xx} < 0 suggests a local maximum. In our example, since D = 3 and f_{xx} = 2, the point (2, 4) is confirmed as a local minimum—a point where the function values are lower than those at nearby points.

Relative extrema are essentially the 'highs' and 'lows' of a function on a given interval or domain. In the context of multivariable functions, these concepts include local maxima, local minima, and saddle points—each one representing a peak, trough, or neither, respectively.

Once we have identified a critical point using the second derivative test, we seek to find the value of the function at that point to confirm the relative extrema. For the critical point (2, 4) in our function, we substitute the coordinates into the original function to find that the local minimum value is -3.

This insight into the relative extrema provides a clear understanding of the function's landscape, revealing where the function's output is greater or less than all nearby outputs. Understanding these points within a function's domain is critical for applications across physics, engineering, and economics, where such extremal values often correspond to optimized conditions.