## 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.