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In linear programming, the objective function is a mathematical representation of the goal that needs to be achieved, typically optimizing a certain quantity of interest. In the context of our exercise, the objective formulates a relationship between two variables, represented by the equation \( P = ax + by \), where \( a \) and \( b \) are positive constants, and \( x \) and \( y \) are the variables whose values we want to optimize.

The values that the objective function can take are dictated by the constraints of the problem. The exercise demonstrates how the value of the function changes when evaluated at different points within the feasible set. A noteworthy point here is that the objective function, as the name suggests, should align with what is being maximized or minimized (e.g., profit, cost, distance). Understanding the objective function is crucial for solving linear programming problems as it directs the search for the optimal solution.

In a linear programming problem, the feasible set is simply the collection of all possible points that satisfy the problem's constraints. Constraints can include inequalities or equations that the variables must abide by. These constraints graphically form a region, and any solution that falls within this region is considered feasible.

As indicated in the exercise, point \( Q \) is a part of the feasible set \( S \) situated on the line segment joining the vertices \( A \) and \( B \), within the bounds defined by the conditions \( 0 < t < 1 \). It is essential for students to realize that not all points within the feasible set are optimal; however, the feasible set provides the boundaries within which the search for an optimal solution is conducted. It is equally important to visualize or sketch this set to have a better understanding of where an optimal solution might lie.

An optimal solution to a linear programming problem is the point or set of points within the feasible set that yield the best possible value of the objective function, adhering to all the problem's constraints. Through the steps provided in the exercise, we can understand that unless the problem allows for infinitely many solutions lying along the line segment that includes \( Q \), then \( Q \) itself cannot be the optimal solution.

Moreover, if the optimal solution lies at a boundary of the feasible set, as is often the case, it is typically found at a vertex or along an edge. However, in some instances, there may be multiple or even infinite optimal solutions – which occurs when the objective function's contours run parallel to a constraint boundary within the feasible set. A deep understanding of the problem's feasible set and objective function shape the path towards identifying where the optimal solution can be located.