extreme f(x,y)=ln(4-x^2-y^2)-sqrt(y)

解答

极值点

f (x, y) = ln (4 - x^{2} - y^{2}) - y

鞍点 (0, 16.48180 \dots)

求解步骤

找到临界点:

(0, 16.48180 \dots)

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{2 ( 4 - x ^{2} + y ^{2} )}{( 4 - x ^{2} - y ^{2} ) ^{2}} + \frac{1}{4 y ^{\frac{3}{2}}}

D (x, y) = \frac{y ^{6} + x ^{2} y ^{4} - 12 y ^{4} - x ^{4} y ^{2} - 8 x ^{2} y ^{2} + 48 y ^{2} - 8 y ^{\frac{11}{2}} - 8 x ^{4} y ^{\frac{3}{2}} - 16 x ^{2} y ^{\frac{7}{2}} + 128 y ^{\frac{3}{2}} - x ^{6} + 4 x ^{4} + 16 x ^{2} - 64}{2 y ^{\frac{3}{2}} ( 4 - x ^{2} - y ^{2} ) ^{4}}

检查在每个临界点处的符号

检查临界点

(0, 16.48180 \dots) :

鞍点

鞍点 (0, 16.48180 \dots)

e x t re m e f (x, y) = x^{2} + 2 x y - y^{2}

e x t re m e f (x, y) = x^{2} y - 5 x y^{3}

e x t re m e f (x, y) = 4 x y^{2} - 2 x^{2} y - x

e x t re m e f (x, y) = 14 x^{2} - 2 x^{3} + 4 x y

e x t re m e f (x) = \frac{1}{3} x^{3} - 5 x^{2} + 21 x - 16