Monotónikus

Monotónikus is a concept often encountered in mathematics, particularly in calculus and analysis. It describes functions that exhibit a consistent directional trend. A function is considered monotonically increasing if for any two values x1 and x2 in its domain, where x1 is less than x2, the value of the function at x1 is less than or equal to the value of the function at x2. In simpler terms, as the input increases, the output either stays the same or increases.

Conversely, a function is monotonically decreasing if for any two values x1 and x2 in its domain, where x1 is less than x2, the value of the function at x1 is greater than or equal to the value of the function at x2. This means that as the input increases, the output either stays the same or decreases.

If a function is strictly monotonically increasing, then for x1 < x2, f(x1) < f(x2). Similarly, for a strictly monotonically decreasing function, x1 < x2 implies f(x1) > f(x2). Functions that are neither monotonically increasing nor decreasing are considered non-monotonic. They may increase in some intervals and decrease in others. The concept of monotonicity is fundamental for understanding function behavior, solving inequalities, and proving theorems related to limits and derivatives.