MHB What Is the Maximum Value of \( x \) in the Equation \( 2x+y+\frac{4}{xy}=10 \)?

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The discussion centers on finding the maximum value of \( x \) in the equation \( 2x+y+\frac{4}{xy}=10 \) for positive real numbers \( x \) and \( y \). Participants are encouraged to share solutions for both current and past Problems of the Week (POTW). One user confirms that posting answers for old POTWs is acceptable and shares their solution along with a plot of the equation. Another participant also contributes an alternative solution. The focus remains on solving the equation effectively to determine the maximum \( x \).
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Here is this week's POTW:

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If the positive real numbers $x$ and $y$ satisfy the equation $2x+y+\dfrac{4}{xy}=10$, find the maximum possible value of $x$.

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No one answered last POTW correctly. Participants can refer to the suggested solution below:

From the given equation, we rewrite it to get

$5-x=\dfrac{y}{2}+\dfrac{2}{xy}>0$ and this implies $x<5$.

Squaring both sides we get

$(5-x)^2=\dfrac{y^2}{4}+\dfrac{2}{a}+\dfrac{4}{x^2y^2}=\dfrac{4}{a}+\left(\dfrac{y}{2}-\dfrac{2}{xy}\right)^2\ge\dfrac{4}{x}$

And so $x(x-5)^2\ge 4$, this translates to $(x-4)(x^2-6x+1)\ge 0$, which, considering $x<5$, implies $x\le 4$.
 
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Are we still allowed to post an answer for an old POTW?
 
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bob012345 said:
Are we still allowed to post an answer for an old POTW?
Of course! I don't see why not, and as a matter of fact, thanks for showing interest in the old POTW!
 
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Likes bob012345, Euge, malawi_glenn and 1 other person
Solve for ##y##:
We get ##y^2 + 2(x-5)y + \dfrac{4}{x}=0##
## y = 5-x \pm \sqrt{(x-5)^2- \dfrac{4}{x}}##
Now we study the discriminant ##d(x) = (x-5)^2- \dfrac{4}{x}##
We have one root ##x=4##, we then set ##d(x) = 0## and multiply both sides with ##x##:
## 0 = x^3-10x^2+25x - 4 ##
This third degree polynomial has ##x=4## as root, the other two roots we find by polymomial long-division:
## (x^3-10x^2+25x - 4)/(x-4) = x^2 - 6x+1 ##
Roots of ## x^2 - 6x+1 ## are ##3-2\sqrt{2}## and ##3+2\sqrt{2}##.
Sketch of the discriminant ##d(x) = (x-5)^2- \dfrac{4}{x}## which we did using a sign-table:
1660929601075.png

Thus in the discriminant, we must have ##3-2\sqrt{2} \leq x \leq 4## or ##x \geq 3+2\sqrt{2}##.
But since ##x \geq 3+2\sqrt{2} > 5## and ## y = 5-x \pm \sqrt{(x-5)^2- \dfrac{4}{x}}## this gives negative ##y##.
Therefore, we must have that ##3-2\sqrt{2} \leq x \leq 4##.

Answer: the largest ##x## value is 4.
 
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Here is my solution and a plot of the equation. I got the same range for ##x## as @malawi_glenn.

Here is my plot of the original equation formatted as ##y= f(x)##, actually the sum of two solutions of the quadratic solution for ##y## and a function of ##x##.

$$ 2x + y + \frac{4}{xy} =10$$
$$ y = \left( \frac{1}{2}\left(10 - 2x\right) +\frac{1}{2} \sqrt{(10-2x)^2 -\frac{16}{x} } \large \right)$$

$$ y = \left( \frac{1}{2}\left(10 - 2x\right) -\frac{1}{2} \sqrt{(10-2x)^2 -\frac{16}{x} } \large \right)$$
The green areas are where ##y## becomes a complex number and purple areas where ##y## is negative.

You can see the range of ##y## is exactly twice the range of ##x## as can be surmised from the symmetry of the equation.

desmos-graph (20) copy.png
 
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Another solution:

Multiply through by ##xy## and we have:
$$2x^2y + xy^2 + 4 - 10xy = 0$$Fix ##x## and let the left-hand side be ##f(y)##.$$f'(y) = 2x^2 + 2xy - 10x = 2x(x + y - 5)$$And ##f(y)## has a minimum at ##y = 5 - x##, where $$f_{min} = 2x^2(5-x) + x(5-x)^2 - 10x(5-x) + 4$$$$ = -x^3 + 10x^2 - 25x + 4 = -(x-4)(x^2 - 6x + 1)$$If ##f_{min} > 0##, then there is no solution for ##y## for that ##x##. We already have an obvious constraint from the original equation that ##x < 5##, so we need ##3 - 2\sqrt 2 \le x \le 4##. And we can check that ##x = 4, y = 1## is indeed a solution.

Alternatively, having noticed that ##x = 4, y = 1## is a solution and ##x < 5## we can see that ##f_{min} > 0## for ##4 < x < 5##. Hence, there are no solutions for ##x > 4##.
 
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