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4.7 Extreme value of functions of two variables

1.

(a) local minimum $-14$ at $(1,4)$ (b) no local extrema (c) local minimum $-1$ at $(1,1)$ (d) local minimum $-1$ at $(\pm 1,0)$

2.

(a) $\displaystyle{f(x,y) = 1 + x + \frac{1}{2}(x^2 - y^2)}$

(b) $\displaystyle{\log{3} + \frac{1}{3}(x-2) + \frac{2}{3}(y-1) - \frac{1}{18}(x-2)^2 - \frac{2}{9}(x-2)(y-1) + \frac{2}{9}(y-1)^2 + \cdots }$