extreme F(x,y)=(1+x^2+y^2)/(1-x^2-y^2)

解答

极值点

F (x, y) = \frac{1 + x ^{2} + y ^{2}}{1 - x ^{2} - y ^{2}}

极小值 (0, 0)

求解步骤

找到临界点:

(0, 0)

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

D (x, y) = \frac{- 48 x ^{4} - 96 x ^{2} y ^{2} + 32 x ^{2} - 48 y ^{4} + 32 y ^{2} + 16}{( - x ^{2} - y ^{2} + 1 ) ^{6}}

检查在每个临界点处的符号

检查临界点

(0, 0) :

极小值

极小值 (0, 0)

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

e x t re m e f (x, y) = y \cdot ln (x)

e x t re m e f (x) = 3 x^{3} - 3 x

e x t re m e f (x) = 5 x + 10 sin (x)

e x t re m e f (x, y) = - 8 x^{3} y + 2 x y^{2} - 5 x^{2} y^{5} + 12 x