If a, b, x, y are positive numbers such that a+b+x+y=5, then the maximum possible value of (a+b)

step 1 In this problem, let the value of A plus B is equal to capital A. The value of X plus Y is equal

If a, b, x, y are positive numbers such that a+b+x+y=5, then...

If $\mathrm{a}, \mathrm{b}, \mathrm{x}, \mathrm{y}$ are positive numbers such that $\mathrm{a}+\mathrm{b}+\mathrm{x}+\mathrm{y}=5$, then the maximum possible value of $(\mathrm{a}+\mathrm{b})(\mathrm{x}+\mathrm{y})$ is
(a) $\frac{5}{2}$
(b) 5
(c) $\frac{25}{4}$
(d) $\frac{25}{2}$

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Key Concepts

AM-GM Inequality

The arithmetic mean-geometric mean (AM-GM) inequality is a principle stating that for any set of non-negative numbers, the arithmetic mean is always at least as great as the geometric mean, with equality holding if and only if all the numbers are equal. This inequality is frequently used in problems where one needs to maximize or minimize the product of numbers given a fixed sum.

Optimization Under Constraints

Optimization under constraints involves finding the maximum or minimum value of an expression subject to certain conditions, such as a fixed sum of variables. This type of problem often requires leveraging equal distribution among the variables, as equal partitioning can yield the optimum result, particularly when dealing with symmetric expressions.

Quadratic Optimization

When a problem can be reduced to a single variable scenario, the resulting expression often takes the form of a quadratic function. The properties of quadratic functions, especially the location of the vertex, can then be used to determine the maximum or minimum value of the expression, thereby solving the optimization problem efficiently.

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