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受欢迎的函数 >

extreme f(x,y)=4x^2+9y^2-2x-18xy^2

解答

极值点

f (x, y) = 4 x^{2} + 9 y^{2} - 2 x - 18 x y^{2}

解答

极小值 (\frac{1}{4}, 0), 鞍点 (\frac{1}{2}, \frac{1}{3}), 鞍点 (\frac{1}{2}, - \frac{1}{3})

求解步骤

找到临界点:

(\frac{1}{4}, 0), (\frac{1}{2}, \frac{1}{3}), (\frac{1}{2}, - \frac{1}{3})

\frac{\partial ^{2} f}{\partial x ^{2}} = 18 - 36 x

D (x, y) = 8 (18 - 36 x) - 1296 y^{2}

检查在每个临界点处的符号

检查临界点

(\frac{1}{4}, 0) :

极小值

检查临界点

(\frac{1}{2}, \frac{1}{3}) :

鞍点

检查临界点

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

鞍点

极小值 (\frac{1}{4}, 0), 鞍点 (\frac{1}{2}, \frac{1}{3}), 鞍点 (\frac{1}{2}, - \frac{1}{3})

作图

流行的例子

extreme f(x)=(x^3)/3-10x^2+96x+100

e x t re m e f (x) = \frac{x ^{3}}{3} - 10 x^{2} + 96 x + 100

extreme f(x)=-x^3+9x^2+120x-200,x>= 5

e x t re m e f (x) = - x^{3} + 9 x^{2} + 120 x - 200, x \geq 5

extreme f(x)=x^{3/4}-2x^{3/4}

e x t re m e f (x) = x^{\frac{3}{4}} - 2 x^{\frac{3}{4}}

extreme f(x)=x^{2/3}(x-6)

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

extreme (16a^4+16a)/(a^2+1-a)

e x t re m e \frac{16 a ^{4} + 16 a}{a ^{2} + 1 - a}