In Exercises 1–4, use the steps shown in Exercise 1 to find...

In Exercises 1–4, use the steps shown in Exercise 1 to find non- negative numbers x and y that satisfy the given requirements. Give the optimum value of the indicated expression.
$x+y=90$ and $x^{2} y$ is maximized.

Key Concepts

Feasible Region

The feasible region is the set of all possible values of the variables that satisfy the constraints of the problem. In optimization problems, solutions must lie within this region, which might be defined by equations, inequalities, or other conditions.

Substitution Method

The substitution method is a technique used to reduce the number of variables in an optimization problem by expressing one variable in terms of others using the constraint equations. This simplification allows the problem to be solved more easily, typically transforming it into a single-variable problem.

Differentiation in Optimization

Differentiation is used in optimization to find the critical points of a function by taking its derivative and setting the derivative equal to zero. These critical points are then tested to determine whether they yield a maximum or minimum value of the objective function.

Constrained Optimization

This concept involves finding the maximum or minimum of an objective function under given restrictions or constraints. The constraints limit the values that the variables can take, and the solution must satisfy these conditions while optimizing the function.

Objective Function

An objective function is the function that needs to be maximized or minimized in an optimization problem. In many problems, this function represents a quantity of interest such as profit, area, volume, or other measurable outcomes.

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