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

解答

极值点

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

极大值 (- \frac{1}{2}, \frac{1}{6})

求解步骤

找到临界点:

(- \frac{1}{2}, \frac{1}{6})

\frac{\partial ^{2} f}{\partial x ^{2}} = e^{y - x^{2} - 3 y^{2} - x} (1 - 6 y + 6) (1 - 6 y - 6)

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

检查在每个临界点处的符号

检查临界点

(- \frac{1}{2}, \frac{1}{6}) :

极大值

极大值 (- \frac{1}{2}, \frac{1}{6})

e x t re m e \frac{π}{24} sec^{2} (\frac{π x}{24})

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

e x t re m e f (x) = x^{3} - 2 x^{2} - 15 x + 5 (- 2)

e x t re m e f (x) = x^{2} - 18 ln (x)

e x t re m e f (x) = 4 cos (2 x) + 1