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受欢迎的函数 >

extreme f(x,y)=7(x-y)e^{-x^2-y^2}

解答

极值点

f (x, y) = 7 (x - y) e^{- x^{2} - y^{2}}

解答

极小值 (- \frac{1}{2}, \frac{1}{2}), 极大值 (\frac{1}{2}, - \frac{1}{2})

求解步骤

找到临界点:

(- \frac{1}{2}, \frac{1}{2}), (\frac{1}{2}, - \frac{1}{2})

\frac{\partial ^{2} f}{\partial x ^{2}} = 7 (- 4 e^{- x^{2} - y^{2}} y^{3} + 4 e^{- x^{2} - y^{2}} x y^{2} + 6 e^{- x^{2} - y^{2}} y - 2 e^{- x^{2} - y^{2}} x)

D (x, y) = 784 e^{- 2 x^{2} - 2 y^{2}} x^{4} y^{2} - 784 e^{2 (- x^{2} - y^{2})} x^{4} y^{2} - 392 e^{- 2 x^{2} - 2 y^{2}} x^{4} - 1568 e^{- 2 x^{2} - 2 y^{2}} x^{3} y^{3} + 1568 e^{2 (- x^{2} - y^{2})} x^{3} y^{3} + 784 e^{- 2 x^{2} - 2 y^{2}} x^{3} y + 784 e^{- 2 x^{2} - 2 y^{2}} x^{2} y^{4} - 784 e^{2 (- x^{2} - y^{2})} x^{2} y^{4} - 784 e^{- 2 x^{2} - 2 y^{2}} x^{2} y^{2} + 588 e^{- 2 x^{2} - 2 y^{2}} x^{2} - 196 e^{2 (- x^{2} - y^{2})} x^{2} + 784 e^{- 2 x^{2} - 2 y^{2}} x y^{3} - 1568 e^{- 2 x^{2} - 2 y^{2}} x y - 392 e^{- 2 x^{2} - 2 y^{2}} y^{4} + 588 e^{- 2 x^{2} - 2 y^{2}} y^{2} - 196 e^{2 (- x^{2} - y^{2})} y^{2}

检查在每个临界点处的符号

检查临界点

(- \frac{1}{2}, \frac{1}{2}) :

极小值

检查临界点

(\frac{1}{2}, - \frac{1}{2}) :

极大值

极小值 (- \frac{1}{2}, \frac{1}{2}), 极大值 (\frac{1}{2}, - \frac{1}{2})

作图

流行的例子

extreme y=x^2-8x

e x t re m e y = x^{2} - 8 x

extreme 2cos(x)+cos^2(x)

e x t re m e 2 cos (x) + cos^{2} (x)

extreme x^{19/9}+x^{10/9}

e x t re m e x^{\frac{19}{9}} + x^{\frac{10}{9}}

extreme-10x^4+4x^2-6

e x t re m e - 10 x^{4} + 4 x^{2} - 6

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

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