4.7 Extreme values of functions of two variables

1.

(a) $${1 + 2(x-1) + y-1 + (x-1)^{2} + 2(x-1)(y-1)}$$

=2.6zw =1(b) $${-\frac{\pi}{2}(x-1) - (y - \frac{\pi}{2}) + \frac{1}{2}\left(-\fr... ...^{2}}{4}(x-1)^{2} - 2(x-1)(y - \frac{\pi}{2}) - (y - \frac{\pi}{2})^{2}\right)}$$

(c) $${-(x-1) + y-1 + \frac{1}{2}\left(-(x-1)^{2} + 2(x-1)(y-1) - (y-1)^{2}\right)}$$

(d) $${x + \frac{1}{2}(4x^{2} + 2xy)}$$

2.

(a) $${f(1,0) = 1}$$ maximum (b) $${f(0,4) = -32}$$ minimum, $f(0,0) = 0$ saddle point　

(c) $${f(1,0) = -2}$$ minimum, $f(-1,0) = 4$ saddle point (d) $${f(1,1) = -3}$$ minimum