extreme f(x)=(xy+6)/(3sqrt(x^2-y))

解答

极值点

f (x) = \frac{x y + 6}{3 x ^{2} - y}

鞍点 (- \frac{3 ^{\frac{1}{3}}}{2 ^{\frac{1}{3}}}, - 318)

求解步骤

找到临界点:

(- \frac{3 ^{\frac{1}{3}}}{2 ^{\frac{1}{3}}}, - 318)

\frac{\partial ^{2} f}{\partial x ^{2}} = \frac{4 x ^{3} - x y + 18}{12 ( x ^{2} - y ) ^{\frac{5}{2}}}

D (x, y) = \frac{48 x ^{5} - 4 x ^{4} y ^{2} - 132 x ^{3} y + 5 x ^{2} y ^{3} - 108 x ^{2} + 84 x y ^{2} - y ^{4} + 108 y}{36 ( x ^{2} - y ) ^{5}}

检查在每个临界点处的符号

检查临界点

(- \frac{3 ^{\frac{1}{3}}}{2 ^{\frac{1}{3}}}, - 318) :

鞍点

鞍点 (- \frac{3 ^{\frac{1}{3}}}{2 ^{\frac{1}{3}}}, - 318)

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

e x t re m e f (x) = \frac{1}{2} x^{4} - x^{2} - 1

e x t re m e f (x) = 3 + 9 x^{2} - 6 x^{3}

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

e x t re m e f (x) = a r cos (1 - ∣ x ∣) - π