<Question>
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz(x + y + z) = 1.$  Find the minimum value of
\[(x + y)(y + z).\]
</Question>

<think>
<Plan_1>
Okay, so I need to find the minimum value of (x + y)(y + z) given that x, y, z are positive real numbers and they satisfy the condition xyz(x + y + z) = 1. Hmm, let's start by understanding what's given and what's required.

First, the variables are all positive, which is good because it means I don't have to worry about negative values messing things up. The constraint is xyz times the sum of x, y, z equals 1. And the expression to minimize is the product of (x + y) and (y + z). 

Let me write down the expression and the constraint to have them clear:

Expression to minimize: (x + y)(y + z)

Constraint: xyz(x + y + z) = 1

I wonder if there's a way to apply some inequality here, like AM-GM or Cauchy-Schwarz. Those are common techniques for optimization problems with constraints.

Let me expand the expression to see if that helps. Maybe expanding will make it easier to handle.

Expanding (x + y)(y + z):

<prompt_1.1> Expand the expression (x + y)(y + z) and consider if it can be related to the constraint. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
= xy + xz + y² + yz

Hmm, so that's the expanded form. Now, the constraint is xyz(x + y + z) = 1. Let's denote S = x + y + z. Then the constraint becomes xyzS = 1. 

I need to relate the expression we need to minimize to this constraint. Maybe if I can express some variables in terms of others or find some symmetry.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Since the expression to minimize is (x + y)(y + z) and the constraint involves x, y, z, maybe symmetry can be used here. If I assume x = z, perhaps the problem becomes symmetric in x and z, which might simplify things. Let me try that.

Let’s set x = z. Then, the constraint becomes x * y * x * (x + y + x) = x²y(2x + y) = 1.

<prompt_2.1> Assume x = z and simplify the expression to minimize. </prompt_2.1>
-----

Let me check another case where maybe y is different. Suppose, for example, that y is not equal to x or z. Maybe if I set x = z, but let y vary. Wait, but in the case above, when I set x = z, the expression simplified nicely, and the minimum was found at x = z = y. So maybe that's the only case.

Alternatively, suppose I don't assume x = z. Let's try another approach.

Let me use Lagrange multipliers. Let’s set up the problem:

<prompt_2.2> Use Lagrange multipliers to find the minimum value without assuming x = z. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
So that's x²y(2x + y) = 1. And the expression to minimize becomes (x + y)(y + x) = (x + y)². Wait, no. Wait, (x + y)(y + z) becomes (x + y)(y + x) since z = x. So that's (x + y)².
</execution_2.1>

<execution_2.2>
Minimize f(x, y, z) = (x + y)(y + z)

Subject to the constraint g(x, y, z) = xyz(x + y + z) - 1 = 0

The Lagrangian is L = (x + y)(y + z) - λ[xyz(x + y + z) - 1]

Compute the partial derivatives:

∂L/∂x = (y + z) - λ[yz(x + y + z) + xyz] = 0

∂L/∂y = (x + y) + (y + z) - λ[xz(x + y + z) + xyz] = 0

∂L/∂z = (x + y) - λ[xy(x + y + z) + xyz] = 0

And the constraint: xyz(x + y + z) = 1

This seems complicated, but maybe we can find some symmetry here. Let's suppose that x = z, as before. Then, perhaps the system simplifies.

Let x = z = a, then y = b. Then the constraint becomes a * b * a * (a + b + a) = a²b(2a + b) = 1.

The function to minimize is (a + b)^2. Wait, no: (x + y)(y + z) = (a + b)(b + a) = (a + b)^2. So we need to minimize (a + b)^2 given that a²b(2a + b) = 1.

Let me set t = a/b. Then a = tb. Substitute into the constraint:

(t b)^2 * b * (2 t b + b) = t² b³ * b (2t + 1) = t² (2t + 1) b^4 = 1

So b^4 = 1 / [t² (2t + 1)]

Thus, b = [1 / (t² (2t + 1))]^{1/4}

Then, (a + b)^2 = (tb + b)^2 = b²(t + 1)^2 = [1 / (t² (2t + 1))]^{1/2} * (t + 1)^2

So the expression to minimize is [(t + 1)^2] / sqrt(t² (2t + 1))

Let’s denote this as E(t) = (t + 1)^2 / [t sqrt(2t + 1)]

We need to find the minimum of E(t) for t > 0.

Take derivative of E(t) with respect to t:

Let’s compute dE/dt. Let me write E(t) as (t + 1)^2 / [t (2t + 1)^{1/2}]

Let’s use logarithmic differentiation. Let’s take ln E(t) = 2 ln(t + 1) - ln t - (1/2) ln(2t + 1)

Then, derivative is [2/(t + 1)] - [1/t] - [ (1/2)*(2/(2t + 1)) ] = 2/(t + 1) - 1/t - 1/(2t + 1)

Set derivative to zero:

2/(t + 1) - 1/t - 1/(2t + 1) = 0

Multiply all terms by t(t + 1)(2t + 1) to eliminate denominators:

2t(2t + 1) - (t + 1)(2t + 1) - t(t + 1) = 0

Expand each term:

First term: 2t(2t + 1) = 4t² + 2t

Second term: -(t + 1)(2t + 1) = -(2t² + t + 2t + 1) = -2t² -3t -1

Third term: -t(t + 1) = -t² - t

Combine all terms:

4t² + 2t -2t² -3t -1 -t² -t = (4t² -2t² -t²) + (2t -3t -t) + (-1) = t² -2t -1 = 0

So quadratic equation: t² -2t -1 = 0

Solutions: t = [2 ± sqrt(4 + 4)] / 2 = [2 ± sqrt(8)] / 2 = [2 ± 2*sqrt(2)] / 2 = 1 ± sqrt(2)

Since t > 0, we take t = 1 + sqrt(2)

Therefore, the minimum occurs at t = 1 + sqrt(2)

Now, compute E(t) at this t:

E(t) = (t + 1)^2 / [t sqrt(2t + 1)]

Let’s compute t + 1 = 1 + sqrt(2) + 1 = 2 + sqrt(2)

2t + 1 = 2*(1 + sqrt(2)) + 1 = 2 + 2sqrt(2) + 1 = 3 + 2sqrt(2) = (sqrt(2) + 1)^2

Therefore, sqrt(2t + 1) = sqrt( (sqrt(2) + 1)^2 ) = sqrt(2) + 1

Thus, E(t) = (2 + sqrt(2))² / [ (1 + sqrt(2)) (sqrt(2) + 1) ) ]

Wait, denominator is t sqrt(2t + 1) = (1 + sqrt(2))(sqrt(2) + 1) = (1 + sqrt(2))²

Numerator is (2 + sqrt(2))²

So E(t) = (2 + sqrt(2))² / (1 + sqrt(2))²

But note that 2 + sqrt(2) = sqrt(2)*(sqrt(2)) + sqrt(2) = sqrt(2)(sqrt(2) + 1)

Wait, let's compute:

(2 + sqrt(2)) = sqrt(2)*(sqrt(2)) + sqrt(2) = sqrt(2)*(sqrt(2) + 1)

Therefore, (2 + sqrt(2))² = [sqrt(2)(1 + sqrt(2))]² = 2*(1 + sqrt(2))²

Thus, E(t) = [2*(1 + sqrt(2))²] / (1 + sqrt(2))² = 2

So the minimum value of (x + y)(y + z) is 2.

But wait, this is under the assumption that x = z. But earlier, when we assumed x = z and found the minimum at x = z = y, we also got 2. So this seems consistent. Therefore, even without assuming x = z, the minimum is 2, achieved when x = z = y.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Let me denote S = x + y + z, so the constraint is xyzS = 1. Then, we need to minimize (x + y)². But (x + y)² = x² + 2xy + y². Hmm, not sure if that helps.

Alternatively, maybe express variables in terms of two variables. Let's say, set x = z, as before, but maybe that's too restrictive. Let's try another substitution.

Alternatively, let's use AM-GM on the expression (x + y)(y + z). Let's expand it again: xy + xz + y² + yz = y(x + y + z) + xz. Wait, that's interesting. Because x + y + z is S, so yS + xz. So (x + y)(y + z) = yS + xz.

<prompt_3.1> Use AM-GM inequality to find a lower bound for the expression. </prompt_3.1>
-----

Based on execution_2.2:
But let me check if this is indeed the case. Let me verify if when x = y = z, the constraint is satisfied.

<prompt_3.2> Verify if the solution satisfies the constraint and conclude the minimum value. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
But from the constraint, xyzS = 1. So, since xyzS = 1, then xz = 1/(yS). Therefore, substituting back into the expression:

(x + y)(y + z) = yS + 1/(yS)

So now, the expression to minimize is yS + 1/(yS). That seems like something we can apply AM-GM on. Because for positive numbers a and b, a + 1/a >= 2, with equality when a = 1. So here, if we let a = yS, then yS + 1/(yS) >= 2. So the minimum value would be 2, achieved when yS = 1. Therefore, (x + y)(y + z) >= 2, with equality when yS = 1 and x = z.

Wait, so does that mean the minimum is 2? But we need to check if equality can be achieved. That is, when yS = 1 and x = z. Let's see.

If x = z, then from the constraint xyzS = 1, with x = z, we have x^2 y S = 1. But S = x + y + z = 2x + y. So x^2 y (2x + y) = 1. If we also have yS = 1, then y(2x + y) = 1. Let me write that as 2x y + y² = 1. Let's denote y = t. Then 2x t + t² = 1. Also, from x^2 t (2x + t) = 1. Let's see if we can solve these equations.

From yS = 1: t(2x + t) = 1 => 2x t + t² = 1.

From the constraint: x² t (2x + t) = x² t * (1/t) = x². Because 2x t + t² = 1, so (2x + t) = 1/t. Therefore, x² t (2x + t) = x² t * (1/t) = x² = 1. Therefore, x² = 1 => x = 1 (since x is positive). Then, from 2x t + t² = 1, with x = 1: 2*1*t + t² = 1 => t² + 2t - 1 = 0. Solving this quadratic equation: t = [-2 ± sqrt(4 + 4)] / 2 = [-2 ± sqrt(8)] / 2 = [-2 ± 2*sqrt(2)] / 2 = -1 ± sqrt(2). Since t is positive, t = -1 + sqrt(2). Wait, sqrt(2) is approx 1.414, so -1 + 1.414 ≈ 0.414, which is positive. So t = sqrt(2) - 1.

Therefore, x = z = 1, y = sqrt(2) - 1. Let's check if this satisfies the constraint:

xyzS = 1 * (sqrt(2) - 1) * 1 * (2*1 + (sqrt(2) - 1)) = (sqrt(2) - 1)(2 + sqrt(2) - 1) = (sqrt(2) - 1)(1 + sqrt(2)).

Multiply these terms: (sqrt(2) - 1)(sqrt(2) + 1) = (sqrt(2))^2 - 1^2 = 2 - 1 = 1. So yes, the constraint is satisfied.

But wait, earlier when I assumed x = y = z, we got a different result. Let me check that again. If x = y = z = a, then the constraint becomes a * a * a * (3a) = 3a^4 = 1 => a = (1/3)^{1/4}. Then (x + y)(y + z) = (2a)(2a) = 4a² = 4*(1/3)^{1/2} = 4/sqrt(3) ≈ 2.309, which is larger than 2. So in that case, the minimum when variables are equal is higher than 2, so the earlier approach where x = z and y different gives a lower value. Therefore, the minimum is indeed 2, achieved when x = z = 1, y = sqrt(2) - 1.
</execution_3.1>

<execution_3.2>
If x = y = z = a, then the constraint becomes a * a * a * (a + a + a) = a³ * 3a = 3a⁴ = 1. So a⁴ = 1/3, so a = (1/3)^{1/4} = 3^{-1/4}. Then, the expression (x + y)(y + z) = (a + a)(a + a) = (2a)(2a) = 4a². Plugging in a = 3^{-1/4}, we get 4 * 3^{-1/2} = 4 / sqrt(3) ≈ 2.309, which is larger than 2. So this contradicts the previous result. Wait, that can't be. So if we set x = z, but not necessarily equal to y, we get a lower value. So maybe the minimum is indeed 2, achieved when x = z but y is different.

Wait, but in the case where x = z = a and y = b, with a ≠ b, we found that the minimum of (a + b)^2 is 2, achieved when t = 1 + sqrt(2). So that suggests that the minimum is 2. But when we set x = y = z, we get a higher value. Therefore, the minimum is indeed 2, and it occurs when two variables are equal (x = z) and the third is different (y).

But wait, in the Lagrangian approach, we didn't assume x = z, but through the process, we found that the minimum occurs at t = 1 + sqrt(2), which corresponds to x = z. So the minimum is 2, achieved when x = z and y is such that t = (x/y) = 1 + sqrt(2). Therefore, x = (1 + sqrt(2)) y, and z = x. So all variables are related through this ratio.

But let me check again.

Wait, when we set x = z = a, y = b, then the expression (x + y)(y + z) = (a + b)^2. The constraint is a²b(2a + b) = 1. Then, by setting t = a/b, we found that the minimum of (a + b)^2 is 2, achieved when t = 1 + sqrt(2). Therefore, the minimum value is 2, which is less than the case when x = y = z. Therefore, the answer should be 2.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1:
But let me verify this conclusion again. Because when I first approached the problem, I assumed x = z and found the minimum at x = z = y, giving 2. But then, when I considered x = z but y different, I found that the minimum is still 2, but with different variable values. Wait, actually, in the first case, when I set x = z = y, then the expression (x + y)(y + z) = (2x)(2x) = 4x². Let's see what x is in that case.

<prompt_4.1> Verify the conclusion by checking the expression when x = z = y. </prompt_4.1>
-----

Based on execution_3.2:
But let me check once more with different numbers.

Suppose x = z = 1, y = sqrt(2) - 1 ≈ 0.414. Then, (x + y)(y + z) = (1 + 0.414)^2 ≈ (1.414)^2 ≈ 2. Also, xyz(x + y + z) = 1 * 0.414 * 1 * (1 + 0.414 + 1) ≈ 0.414 * 2.414 ≈ 1. Which satisfies the constraint.

Alternatively, if we take x = z = t, y = s, then from the constraint t² s (2t + s) = 1.

<prompt_4.2> Verify the solution by checking the constraint and expression with specific values. </prompt_4.2>
</Plan_4>

<Execution_4>
<execution_4.1>
If x = z = y = a, then the constraint is a * a * a * (a + a + a) = a³ * 3a = 3a⁴ = 1. So a⁴ = 1/3, so a = (1/3)^{1/4}. Then, (x + y)(y + z) = 4a² = 4*(1/3)^{1/2} = 4/sqrt(3) ≈ 2.309. But according to the earlier AM-GM step, we have (x + y)(y + z) = yS + xz. If x = z = 1, y = sqrt(2) - 1, then S = x + y + z = 1 + (sqrt(2) - 1) + 1 = 1 + sqrt(2) - 1 + 1 = 1 + sqrt(2). Then yS = (sqrt(2) - 1)(1 + sqrt(2)) = (sqrt(2))^2 - 1^2 = 2 - 1 = 1. So yS = 1, and xz = 1*1 = 1. Therefore, (x + y)(y + z) = 1 + 1 = 2. So this works. Therefore, even though when x = y = z, the expression is higher, when x = z but y is different, we can achieve a lower value of 2. Therefore, the minimum is indeed 2.
</execution_4.1>

<execution_4.2>
We can express s in terms of t or vice versa. From the expression (x + y)(y + z) = (t + s)^2. If we set t = (1 + sqrt(2)) s, as per the earlier substitution, then t = (1 + sqrt(2)) s, so s = t / (1 + sqrt(2)). Then, substitute into the constraint:

t² * (t / (1 + sqrt(2))) * (2t + t / (1 + sqrt(2))) = 1

Simplify:

t³ / (1 + sqrt(2)) * (2t + t / (1 + sqrt(2))) = 1

Factor t:

t³ / (1 + sqrt(2)) * t (2 + 1 / (1 + sqrt(2))) = 1

Multiply the terms:

t^4 / (1 + sqrt(2)) * (2 + 1 / (1 + sqrt(2))) = 1

Compute the coefficient:

Let’s compute 2 + 1 / (1 + sqrt(2)):

Multiply numerator and denominator by (1 - sqrt(2)):

1 / (1 + sqrt(2)) = (1 - sqrt(2)) / (1 - 2) = (sqrt(2) - 1)/1 = sqrt(2) - 1

Thus, 2 + sqrt(2) - 1 = 1 + sqrt(2)

Therefore, the coefficient becomes (1 + sqrt(2)) / (1 + sqrt(2)) = 1. Therefore, t^4 * 1 = 1 => t^4 = 1 => t = 1.

Therefore, t = 1, so s = 1 / (1 + sqrt(2)) = sqrt(2) - 1 (rationalizing the denominator). Therefore, this gives x = z = 1, y = sqrt(2) - 1, which matches our earlier result. Therefore, (x + y)(y + z) = (1 + sqrt(2) - 1)^2 = (sqrt(2))² = 2. So yes, this satisfies the condition.
</execution_4.2>
</Execution_4>

<Plan_5>
Based on execution_4.2:
But let me check if there's any other possible configuration where (x + y)(y + z) could be smaller. For example, if we set y to be very small, then x and z would have to compensate to satisfy the constraint. Let's see.

<prompt_5.1> Conclude the solution by checking if smaller values than 2 are possible. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
Suppose y approaches 0. Then, the constraint xyz(x + y + z) = 1 implies that x and z must approach infinity such that xzS ≈ xz(x + z) ≈ xz*2x = 2x²z ≈ 1. So x²z ≈ 1/(2x). Therefore, xz ≈ 1/(2x²). Then, (x + y)(y + z) ≈ x*z ≈ 1/(2x²). As x approaches infinity, this product approaches zero. Wait, but this contradicts our earlier conclusion. What's happening here?

Wait, no. If y approaches zero, then to satisfy the constraint, xzS must approach infinity, but since S = x + y + z ≈ x + z, so xz(x + z) ≈ 1. Then, (x + y)(y + z) ≈ x*z. So xz ≈ 1/(x + z). But if x and z are both approaching infinity, x + z is also approaching infinity, so xz ≈ 1/(x + z) would imply that xz approaches zero, which contradicts xz(x + z) ≈ 1. Therefore, this seems impossible. Therefore, perhaps my initial thought is wrong.

Wait, let's clarify. If y approaches zero, then the constraint becomes xyzS ≈ 0 * xzS = 0, which cannot be equal to 1. Therefore, y cannot approach zero. Therefore, y must be bounded below by some positive number. Therefore, the earlier conclusion that the minimum is 2 is correct. Because if we try to set y very small, the constraint cannot be satisfied. Therefore, the variables are bounded away from zero, so the minimum is indeed 2.
</execution_5.1>
</Execution_5>

<Final_answer>
Therefore, the minimum value of \[(x + y)(y + z)\] under the given constraint is \boxed{2}.
</Final_answer>
</think>