extreme f(x)=e^{3x^2+8y^2+13}

解答

极值点

f (x) = e^{3 x^{2} + 8 y^{2} + 13}

极小值 (0, 0)

求解步骤

找到临界点:

(0, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = 16 (16 e^{3 x^{2} + 8 y^{2} + 13} y^{2} + e^{3 x^{2} + 8 y^{2} + 13})

D (x, y) = 96 (6 e^{3 x^{2} + 8 y^{2} + 13} x^{2} + e^{3 x^{2} + 8 y^{2} + 13}) (16 e^{3 x^{2} + 8 y^{2} + 13} y^{2} + e^{3 x^{2} + 8 y^{2} + 13}) - 9216 e^{2 (3 x^{2} + 8 y^{2} + 13)} x^{2} y^{2}

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) = - 34 x^{2} + 550 x

e x t re m e f (x) = x ln (9 x), (0, \infty)

e x t re m e - \frac{8 \cdot x}{x ^{2} + 1}

e x t re m e f (x) = x^{2} - y^{2} - 5 x - 4 y

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