Can I use calculus?

Algebra Level 3

If x x and y y are two positive real numbers such that 4 x + 9 y = 60 4x+9y = 60 , find the maximum value of x y \sqrt{xy} .


The answer is 5.00.

Shivam Jadhav by Shivam Jadhav , 22, India

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2 solutions

Noel Lo
Jun 19, 2015

By AM-GM inequality, we see that 4 x + 9 y 2 > = ( 4 x ) ( 9 y ) = 2 ( 3 ) x y = 6 x y \frac{4x+9y}{2} >=\sqrt{(4x)(9y)} = 2(3)\sqrt{xy} = 6\sqrt{xy} so 60 2 > = 6 x y \frac{60}{2} >=6\sqrt{xy} or 6 x y < = 30 6\sqrt{xy} <=30 . Hence x y < = 5 \sqrt{xy} <=\boxed{5} .

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Moderator note:

Good job. Does the answer remains the same if x x and y y are both negative numbers instead?

@In Response to Challenge Master if x x and y y are both negative numbers , then how would 4 x + 9 y = 60 4x+9y = 60 ?

Anand O R - 5 years, 10 months ago
Hyunmin Na
Jun 14, 2015

only positive numbers that work are x=6 and y=4 unless you count x=15 and y=0 which wouldn't be the maximum anyways so squareroot of 24 = 4.89, but the answer needs to be in integers so it is 5

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Moderator note:

You may not assume that x and y must be integers. Why can't they be irrational numbers?

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