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设函数f ( x ) = 2x + \frac { a } { x ^ { 2 } } ( a > 0 )在( 0 , + \infty )内的最小值为f ( x _ { 0 } ) =\newline 12,则x _ { 0 } =\newline

Solution by Steps

step 1

To find the minimum of the function

f(x) = 2x + \frac{a}{x^2}

for x > 0, we use calculus methods

step 2

We take the derivative of

f(x)

with respect to

x

to find the critical points:

f'(x) = 2 - \frac{2a}{x^3}

step 3

Set the derivative equal to zero to find the critical points:

2 - \frac{2a}{x^3} = 0

step 4

Solve for

x

to find the critical points:

x = \sqrt[3]{\frac{a}{1}}

step 5

Substitute the critical point back into

f(x)

to find the minimum value:

f(\sqrt[3]{a}) = 2\sqrt[3]{a} + \frac{a}{(\sqrt[3]{a})^2} = 3\sqrt[3]{a}

step 6

Given that the minimum value of

f(x)

is 12, we set

3\sqrt[3]{a} = 12

and solve for

a

step 7

Solving for

a

gives

a = 64

step 8

Now, we solve for

x_0

using the equation

12 = 2x_0 + \frac{64}{x_0^2}

step 9

Rearrange the equation to form a cubic equation in terms of

x_0

:

x_0^3 - 6x_0^2 + \frac{64}{2} = 0

step 10

Solve the cubic equation to find the value of

x_0

[1] Answer

x_0 = 4

Key Concept

Optimization using calculus and solving cubic equations

Explanation

To find the minimum of a function, we take its derivative, set it to zero, and solve for the critical points. Then we check which of these points gives the minimum value of the function. In this case, we also used the given minimum value to find a specific parameter

a

, and then solved a cubic equation to find

x_0

.

Generate me a similar question

How to find the derivative of the function

g(x) = 3x^2 + \frac{b}{x}

for x > 0?

某公司财务部有2名男员工3名女员工，销售部有4名男员工1名女员工。现要从中选2名男员工，1名女员工组成工作小组，并要求每部门至少有一名员工入选，则这个工作小组的构成方式有（）种。

Solution by Steps

step 1

计算财务部和销售部的男员工组合方式。财务部有2名男员工，销售部有4名男员工。我们需要从中选出2名男员工，并且每个部门至少有一名员工入选。

step 2

计算财务部选1名男员工，销售部选1名男员工的组合方式：

\binom{2}{1} \times \binom{4}{1} = 2 \times 4 = 8

step 3

计算财务部选0名男员工，销售部选2名男员工的组合方式：

\binom{2}{0} \times \binom{4}{2} = 1 \times 6 = 6

step 4

计算财务部选2名男员工，销售部选0名男员工的组合方式：

\binom{2}{2} \times \binom{4}{0} = 1 \times 1 = 1

step 5

由于每个部门至少有一名员工入选，所以排除财务部选2名男员工，销售部选0名男员工的情况。

step 6

计算女员工的组合方式。财务部有3名女员工，销售部有1名女员工。我们需要从中选出1名女员工，并且每个部门至少有一名员工入选。

step 7

计算财务部选1名女员工，销售部选0名女员工的组合方式：

\binom{3}{1} \times \binom{1}{0} = 3 \times 1 = 3

step 8

计算财务部选0名女员工，销售部选1名女员工的组合方式：

\binom{3}{0} \times \binom{1}{1} = 1 \times 1 = 1

step 9

由于每个部门至少有一名员工入选，所以排除财务部选0名女员工，销售部选0名女员工的情况。

step 10

计算总的组合方式：

8 \times 3 + 6 \times 1 = 24 + 6 = 30

Answer

30

Key Concept

组合数学

Explanation

通过计算每个部门的男员工和女员工的组合方式，并确保每个部门至少有一名员工入选，得出总的组合方式。

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