LAW OF MONOTONY FOR SUM: application exercises

The Law of Monotony for Addition states that if two quantities that vary monotonically (i.e., always increase or always decrease) in the same direction are added, the sum will also vary monotonically in that direction.

formal statement

If f(x) and g(x) are functions that vary monotonically in the same direction (both increasing or both decreasing) on ​​an interval [a, b], then the function h(x) = f(x) + g (x) will also vary monotonically in the same direction in the interval [a, b].

Properties

1. If f(x) and g(x) are increasing, then h(x) = f(x) + g(x) is also increasing.
2. If f(x) and g(x) are decreasing, then h(x) = f(x) + g(x) is also decreasing.

Examples

1. If f(x) = x^2 and g(x) = 2x are increasing on the interval [0, ∞), then h(x) = x^2 + 2x is also increasing on that interval.
2. If f(x) = -x and g(x) = -2x are decreasing on the interval [0, ∞), then h(x) = -x - 2x = -3x is also decreasing on that interval.

Applications

The Law of Monotony for Addition is used in various areas, such as:

1. Mathematical analysis: to study the monotony of functions and its relationship with sum.
2. Optimization: to find the maximum or minimum of a function that is the sum of monotone functions.
3. Economics: to analyze the monotony of utility functions and its relationship with the sum of goods and services.