Dated: 30-10-2024

Extrema of `functions`¹ with Two Variables

Absolute Maximum

A function¹ is said to have absolute maximum on the domain¹ \(\mathbb{R}^2\) if there is a point \((x_0, y_0) \in \mathbb{R}^2\) such that

\[f(x_0, y_0) \ge f(x, y) \text{ for all } (x, y) \in \mathbb{R}^2\]

Absolute Minimum

Similarly, absolute minimum exists if

\[f(x_0, y_0) \le f(x, y) \text{ for all } (x, y) \in \mathbb{R}^2\]

Relative or Local Maximum

The function¹ is said to have relative maximum at point \((x_0, y_0)\) if there exists an open disk \(K\) centered at \((x_0, y_0)\), of radius \(r\).

\[f(x_0, y_0) \ge f(x, y) \text{ for all } (x, y) \in K\]

\[\text{where } K = \{(x, y) \in \mathbb{R}^2 : (x - x_0)^2 + (y - y_0)^2 < r^2\}\]

Relative or Local Minimum

Similarly, for relative minimum, we have

\[f(x_0, y_0) \le f(x, y) \text{ for all } (x, y) \in K\]

Extreme Value Theorem

If \(f(x, y)\) is continuous² on a closed and bounded set \(R\) then the function¹ has absolute maximum and absolute minimum on \(R\).
If a function¹ has relative extrema at \((x_0, y_0)\), then

\[f_x(x_0, y_0) = f_y(x_0, y_0) = 0\]

Saddle point

If \(f(x, y)\) has a disk \(K\) centered at \((a, b)\) and there exists points \((x, y)\) in its domain¹ such that \(f(x, y) > f(a, b)\) and \(f(x, y) < f(a, b)\) then this point on surface \(f(a, b, f(a, b))\) is called saddle point.

Extrema of functions1 with Two Variables