Extreme values

Exercise

1.

Find the extreme value of the following functions．

(a)

$${f(x,y) = 2x^2 + y^2 -xy - 7y}$$

(b)

$${f(x,y) = \frac{x}{y^2} + xy }$$

(c)

$${f(x,y) = x^3 + y^3 - 3xy}$$

(d)

$${f(x,y) = (x^2 + y^2)^2 - 2(x^2 - y^2)}$$

2.

Find the Taylor expansion of the following function around $(a,b)$ with the $2$nd degree in $xとy$.

(a)

$${f(x,y) = e^x \cos{y}, \ (a,b) = (0,0)}$$

(b)

$${f(x,y) = \log{(x + y^2)}, \ (a,b) = (2,1)}$$