(b) \( \begin{array}{l} \operatorname{minimize}(x+2)^{3}+y \\ \text { subject to } x \geq 1 \\ y \geq 0\end{array} \)

To solve the optimization problem \[ \begin{array}{l} \text{minimize } (x+2)^{3} + y \\ \text{subject to } x \geq 1 \\ y \geq 0 \end{array} \] we will analyze the objective function and the constraints. 1. **Objective Function**: The function we want to minimize is \( f(x, y) = (x+2)^{3} + y \). 2. **Constraints**: The constraints are \( x \geq 1 \) and \( y \geq 0 \). ### Step 1: Analyze the Objective Function The term \( (x+2)^{3} \) is a cubic function in \( x \). As \( x \) increases, \( (x+2)^{3} \) also increases. Therefore, to minimize \( (x+2)^{3} \), we should choose the smallest possible value of \( x \) that satisfies the constraint \( x \geq 1 \). ### Step 2: Determine the Minimum Value of \( x \) The minimum value of \( x \) that satisfies the constraint is \( x = 1 \). ### Step 3: Substitute \( x = 1 \) into the Objective Function Now, we substitute \( x = 1 \) into the objective function: \[ f(1, y) = (1 + 2)^{3} + y = 3^{3} + y = 27 + y \] ### Step 4: Minimize with Respect to \( y \) Next, we need to minimize \( 27 + y \) subject to the constraint \( y \geq 0 \). The minimum value of \( y \) that satisfies this constraint is \( y = 0 \). ### Step 5: Calculate the Minimum Value of the Objective Function Substituting \( y = 0 \) into the function gives: \[ f(1, 0) = 27 + 0 = 27 \] ### Conclusion The minimum value of the objective function \( (x+2)^{3} + y \) subject to the constraints \( x \geq 1 \) and \( y \geq 0 \) is \[ \boxed{27} \] This occurs at the point \( (x, y) = (1, 0) \).