) Find KKT conditions for the following problem:
Maximize: −𝑥 − 𝑦
subject to: 𝑥
2 + 𝑦 ≥ 2; 𝑥 + 3𝑦 ≥ 4; 𝑥 + 𝑦^4 ≤ 30

The Karush-Kuhn-Tucker (KKT) conditions are necessary conditions for a solution in nonlinear programming to be optimal, given certain regularity conditions. Let's derive the KKT conditions for the given problem.

### Problem Statement:

Maximize:
\[
f(x, y) = -x - y
\]

Subject to:
\[
g_1(x, y) = x^2 + y - 2 \geq 0
\]
\[
g_2(x, y) = x + 3y - 4 \geq 0
\]
\[
g_3(x, y) = 30 - x - y^4 \geq 0
\]

### Step 1: Define the Lagrangian

The Lagrangian function \( \mathcal{L}(x, y, \lambda_1, \lambda_2, \lambda_3) \) is defined as:
\[
\mathcal{L}(x, y, \lambda_1, \lambda_2, \lambda_3) = -x - y + \lambda_1 (x^2 + y - 2) + \lambda_2 (x + 3y - 4) + \lambda_3 (30 - x - y^4)
\]

where \( \lambda_1 \), \( \lambda_2 \), and \( \lambda_3 \) are the Lagrange multipliers associated with the constraints \( g_1(x, y) \), \( g_2(x, y) \), and \( g_3(x, y) \), respectively.

### Step 2: KKT Conditions

The KKT conditions include the following:

1. **Stationarity**: The gradient of the Lagrangian with respect to the decision variables \( x \) and \( y \) must be zero:
   \[
   \frac{\partial \mathcal{L}}{\partial x} = -1 + \lambda_1 \cdot 2x + \lambda_2 - \lambda_3 = 0
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial y} = -1 + \lambda_1 + \lambda_2 \cdot 3 - \lambda_3 \cdot 4y^3 = 0
   \]

2. **Primal Feasibility**: The original constraints must be satisfied:
   \[
   x^2 + y - 2 \geq 0
   \]
   \[
   x + 3y - 4 \geq 0
   \]
   \[
   30 - x - y^4 \geq 0
   \]

3. **Dual Feasibility**: The Lagrange multipliers must be non-negative:
   \[
   \lambda_1 \geq 0, \quad \lambda_2 \geq 0, \quad \lambda_3 \geq 0
   \]

4. **Complementary Slackness**: The product of each Lagrange multiplier and the corresponding constraint must be zero:
   \[
   \lambda_1 \cdot (x^2 + y - 2) = 0
   \]
   \[
   \lambda_2 \cdot (x + 3y - 4) = 0
   \]
   \[
   \lambda_3 \cdot (30 - x - y^4) = 0
   \]

### Summary of KKT Conditions

The KKT conditions for the given problem are:

1. **Stationarity**:
   \[
   -1 + 2\lambda_1 x + \lambda_2 - \lambda_3 = 0
   \]
   \[
   -1 + \lambda_1 + 3\lambda_2 - 4\lambda_3 y^3 = 0
   \]

2. **Primal Feasibility**:
   \[
   x^2 + y - 2 \geq 0
   \]
   \[
   x + 3y - 4 \geq 0
   \]
   \[
   30 - x - y^4 \geq 0
   \]

3. **Dual Feasibility**:
   \[
   \lambda_1 \geq 0, \quad \lambda_2 \geq 0, \quad \lambda_3 \geq 0
   \]

4. **Complementary Slackness**:
   \[
   \lambda_1 \cdot (x^2 + y - 2) = 0
   \]
   \[
   \lambda_2 \cdot (x + 3y - 4) = 0
   \]
   \[
   \lambda_3 \cdot (30 - x - y^4) = 0
   \]

These conditions must be satisfied at an optimal point for \( x \) and \( y \).

Would you like to explore the solution further or see an example? 

### Related Questions:

1. How do you interpret the complementary slackness condition in the context of this problem?
2. What are the necessary conditions for the KKT conditions to guarantee an optimal solution?
3. How would the KKT conditions change if the objective function were different?
4. Can KKT conditions provide insight into whether a solution is a global or local maximum?
5. How would you solve for \( x \), \( y \), and the Lagrange multipliers numerically?

### Tip:
In optimization problems, complementary slackness helps determine whether a constraint is active or inactive at the optimal solution.

Explore the derivation and application of Karush-Kuhn-Tucker (KKT) conditions in nonlinear programming. Learn how these conditions are essential for optimizing objective functions subject to constraints. Includes a detailed example and step-by-step explanation.

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