The question asks which of the options is a valid objective function.

An objective function is a mathematical expression that needs to be maximized or minimized in an optimization problem. It is typically a linear equation involving decision variables (e.g., $x$ and $y$).

Let’s review the options provided in the image:

1. $R(x, y) = 50x + 75y$
   This is a valid objective function because it is a linear combination of $x$ and $y$ that can be maximized or minimized.

2. $r(x, y) > 30x + 25y$
   This is an inequality, which would likely represent a constraint, not an objective function.

3. $x \geq 25$
   This is a constraint, not an objective function.

4. $5x + 3y \leq 150$
   This is also a constraint, not an objective function.

Thus, the valid objective function is the first option:
$R(x, y) = 50x + 75y$.

Would you like further details or have any questions? Here are some related questions to deepen your understanding:

1. What is the role of constraints in an optimization problem?
2. How do you differentiate between an objective function and a constraint?
3. What are some common methods to solve optimization problems with objective functions?
4. Can an objective function be nonlinear, or must it always be linear?
5. How is the feasible region defined in a linear programming problem?

Tip: When solving optimization problems, always start by clearly identifying the objective function and the constraints separately.