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In optimization problems, the objective function is the function that you want to maximize or minimize. It's the central aspect of what you’re trying to accomplish in any optimization scenario. For this particular problem, the objective function is given by\[ P = xy \]This function represents the value we seek to either maximize or minimize, depending on the context of the problem. Here, we're interested in maximizing the product of \(x\) and \(y\) while respecting the resources or conditions imposed by the problem. Understanding the structure of the objective function is critical because it directly influences the steps needed to find the solution. We are tasked with expressing this function in terms of a single variable, which is where constraint equations play a pivotal role.

Constraint equations are what stand between you and the unconstrained maximization or minimization of the objective function. These equations define the conditions or limitations within which the optimization is performed. In this case, the constraint is given by:\[ 10x + y = 100 \]This tells us there is a relationship between \(x\) and \(y\) that must be maintained. By solving this equation for one of the variables (in this case, \(y\)), we can reframe the objective function purely as a function of \(x\), leading to:\[ y = 100 - 10x \]This simplification helps in reducing the complexity of the problem and allows us to handle the objective function more easily by focusing on a single variable. This is essential to facilitate the process of finding optimal values using calculus, as we’ll explore with critical points.

Critical points are where potential maximum or minimum values of a function occur. For the derived objective function:\[ P = 100x - 10x^2 \]We find critical points by taking the derivative of \(P\) concerning \(x\), then solving for when this derivative equals zero or is undefined:\[ \frac{dP}{dx} = 100 - 20x \]Setting the derivative to zero results in the equation:\[ 100 - 20x = 0 \]Solving this gives:\[ x = 5 \]This \(x\)-value is where the function might hit a maximum or minimum. Identifying critical points allows us to further evaluate and determine whether these points actually correspond to an absolute maximum or minimum in the context of the given domain, which takes us to the concept of absolute maximum.

To determine the absolute maximum of a function within a certain interval, you evaluate the objective function at critical points as well as at the boundaries of the domain. In this problem, the domain is stipulated by the constraint equation, which confines \(x\) to:\[ [0, 10] \]Hence we must evaluate the function \(P = 100x - 10x^2\) at three points - the critical point \(x = 5\) and the boundaries \(x = 0\) and \(x = 10\):- \( P(0) = 100(0) - 10(0)^2 = 0 \)- \( P(5) = 100(5) - 10(5)^2 = 250 \)- \( P(10) = 100(10) - 10(10)^2 = 0 \)Comparing these values, the highest value reached within the interval is \(250\), making 250 the absolute maximum for this problem. Recognizing and calculating the absolute maximum gives you a definitive answer to the maximal value attainable under given constraints.