The problem asks us to fill in the blanks in the statement: "If $f(x, y)$ attains a maximum value at a point $(x_0, y_0)$, then $(x_0, y_0)$ is either a(n) \_\_\_\_\_\_\_ point or a(n) \_\_\_\_\_\_\_ point or a(n) \_\_\_\_\_\_\_ point."

1. Problem Description

The problem asks us to fill in the blanks in the statement: "If $f(x, y)$ attains a maximum value at a point $(x_0, y_0)$, then $(x_0, y_0)$ is either a(n) \_\_\_\_\_\_\_ point or a(n) \_\_\_\_\_\_\_ point or a(n) \_\_\_\_\_\_\_ point."

2. Solution Steps

The maximum value of a function $f(x,y)$ can occur at three types of points:

\begin{itemize}

\item Local maximum: A point where the function value is larger than all surrounding points.

\item Absolute (global) maximum: A point where the function value is the largest over the entire domain.

\item Boundary point: If the domain of the function is restricted, the maximum can occur on the boundary of the domain.

\end{itemize}

Also, the derivative might not exist. Therefore, stationary points and singular points are the other locations of a maximum value.

3. Final Answer

If $f(x, y)$ attains a maximum value at a point $(x_0, y_0)$, then $(x_0, y_0)$ is either a(n) local maximum point or a(n) absolute maximum point or a(n) boundary point.