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受欢迎的函数 >

extreme f(x,y)=xe^{-3x^2-3y^2}

解答

极值点

f (x, y) = x e^{- 3 x^{2} - 3 y^{2}}

解答

极小值 (- \frac{1}{6}, 0), 极大值 (\frac{1}{6}, 0), 极小值 (- \frac{1}{6}, 0), 极大值 (\frac{1}{6}, 0)

求解步骤

找到临界点:

(- \frac{1}{6}, 0), (\frac{1}{6}, 0), (- \frac{1}{6}, 0), (\frac{1}{6}, 0)

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

D (x, y) = - 6 x (36 e^{- 3 (x^{2} + y^{2})} x^{3} - 18 e^{- 3 (x^{2} + y^{2})} x) (- 6 e^{- 3 x^{2} - 3 y^{2}} y^{2} + e^{- 3 x^{2} - 3 y^{2}}) - 36 e^{- 6 (x^{2} + y^{2})} y^{2} (6 x + 1)^{2} (6 x - 1)^{2}

检查在每个临界点处的符号

检查临界点

(- \frac{1}{6}, 0) :

极小值

检查临界点

(\frac{1}{6}, 0) :

极大值

检查临界点

(- \frac{1}{6}, 0) :

极小值

检查临界点

(\frac{1}{6}, 0) :

极大值

极小值 (- \frac{1}{6}, 0), 极大值 (\frac{1}{6}, 0), 极小值 (- \frac{1}{6}, 0), 极大值 (\frac{1}{6}, 0)

作图

流行的例子

extreme f(x,y)=2x^2-8x+y^2-8y+2

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

extreme 1/(x-2)

e x t re m e \frac{1}{x - 2}

extreme f(x,y)=x^2+y^2-6y+9

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

extreme f(x)=x^2,-1<= x<= 0

e x t re m e f (x) = x^{2}, - 1 \leq x \leq 0

extreme f(x)=x^4-4x^3+16x+4

e x t re m e f (x) = x^{4} - 4 x^{3} + 16 x + 4