Bounded extrema
Task number: 3233
For the given function \(f\) determine the global extrema of the function \(f\) on the given set \(M\). Verify that the points are really global extrema.
Variant 1
\(f(x, y) = 2x + y\) a \(M = \{(x, y) \in \mathbb R^2\colon \space x^2 + y^2 = 1\}\).
Variant 2
\(f(x, y, z) = x + y + z\) a \(M = \{(x, y, z) \in \mathbb R^3 \colon \space x^2 + y^2 + z^2 = 1\}\)
Variant 3
\(f(x, y) = x^2 - 2x + y^2 - 4y\) a \(M = \{(x, y) \in \mathbb R^2\colon \space x^2 + y^2 = 20 \}\)
Variant 4
\(f(x, y) = x^2 - 2x + y^2 - 4y\) a \(M = \{(x, y) \in \mathbb R^2\colon \space x^2 + y^2 \leq 20 \}\)
Variant 5
\(f(x, y) = x^2 -4 xy + y^2 + 4y\) a \(M = \{(x, y) \in \mathbb R^2\colon \space 0 \leq x \leq 1, \space 0 \leq y \leq 1 \}\)
Variant 6
\(f(x, y) = x^2 + y^2 + z^2\) a \(M = \{(x, y, z) \in \mathbb R^3\colon \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 0 \}\), where \(a, b, c > 0\) are parameters.
