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已知

\mathrm{r}>2, y>1, x y-x-2 y=2

，则

x+y

的最小值是

Mar 8, 2024

已知 \mathrm{r}>2, y>1, x y-x-2 y=2 ，则

x+y

的最小值是

Generated Graph

Solution by Steps

step 1

To find the minimum value of

x + y

given the constraints

xy - x - 2y = 2

, x > 0 , and y > 1 , we can use the method of Lagrange multipliers or solve the system of equations directly

step 2

First, we rearrange the constraint equation to express

y

in terms of

x

y = \frac{x + 2}{x - 2}

step 3

We then substitute this expression for

y

into

x + y

to get a function of

x

alone, which we can minimize

step 4

After substituting, we find the derivative of the resulting function with respect to

x

, set it to zero, and solve for

x

to find the critical points

step 5

We check the critical points to find the minimum value of

x + y

within the domain given by the constraints

step 6

The minimum value of

x + y

is found to be 7 at the point

(x, y) = (4, 3)

Answer

The minimum value of

x + y

is 7.

Key Concept

Optimization with constraints

Explanation

To find the minimum value of a function subject to constraints, we can use methods such as substitution or Lagrange multipliers to reduce the problem to one variable and then find the critical points that satisfy the constraints.