perplexus.info :: Just Math : Given the product of linear expression, find the minimum

Given the product of linear expression, find the minimum (Posted on 2025-03-28)

If x, y>0 such that:
(x+y)(1/x+1/y)=5
Then, find the minimum value of:
(x³+y³)(1/x³+1/y³)

No Solution Yet Submitted by K Sengupta

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different approach,same answer

Comment 2 of 2 |

Rewrite the equation as (x+y)^2/xy=5.

The expression to be minimized can be rearranged and factored as (x^3+y^3)^2/(xy)^3=(x+y)^2*(x^2-xy+y^2)^2/(xy)^3.

Substitute the value from the equation and the expression becomes 5*(x^2-xy+y^2)^2/(xy)^2.

Write this as 5*((x+y)^2-3xy)/xy)^2

Divide each term by xy: 5*((x+y)^2/xy-3))^2

Substitute again: 5*(5-3)^2=5*4=20

Posted by xdog on 2025-03-28 14:14:02

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