Extremum with side conditions

Exercise

1.

Find the extreme of the implicit function $y = g(x)$ derived from the following function．

(a)

$${8x^2 +4xy + 5y^2 = 36}$$

(b)

$${x^{2}y + x + y = 0}$$

(c)

$${x^3 + y^3 - 6xy = 0}$$

2.

Find the maximum and minimum values of $f(x,y)$ under the following condition $g(x,y) = 0$.

(a)

$${g(x,y) = x^2 + y^2 - 1, \ f(x,y) = xy^3}$$

(b)

$${g(x,y) = x^3 + y^3 - 6xy, \ f(x,y) = x^2 + y^2}$$

(c)

$${g(x,y) = x^2 - xy + y^2 - 1, \ f(x,y) = xy}$$

3.

Find the maximum value of $xy$ when the point ${\rm P}(x,y)$ moves on the straight line $2x + 3y = 12$．

4.

When the point ${\rm P}(x,y,z)$ moves over the sphere $x^2+y^2+z^2=1$, Find the maximum and minimum values of $x^2+2y^2+3z^2$.