increaseanddecrease

Increase and decrease describe how the value of a function or sequence changes as the input or index grows. In mathematics, a function f defined on an interval I is increasing if, for any x1 and x2 in I with x1 < x2, we have f(x1) ≤ f(x2). If the inequality is strict for all such pairs, f is strictly increasing. Decreasing is defined analogously: x1 < x2 implies f(x1) ≥ f(x2), and strictly decreasing when the inequality is strict. These properties together form monotonicity, a key concept in analysis.

A function may be increasing on some subintervals and decreasing on others, producing distinct intervals of increase and decrease. Identifying these intervals is common in calculus and real analysis, as they reveal shape, turning points, and potential maxima or minima.

For differentiable functions, the sign of the derivative provides a practical test: if f′(x) > 0 on an interval, then f is strictly increasing there; if f′(x) < 0, then f is strictly decreasing. Points where f′(x) changes sign indicate local extrema. A zero derivative does not automatically determine monotonicity; it may indicate a turning point, a horizontal inflection, or no change in monotonicity at all, depending on the surrounding behavior.

Examples illustrate the concept: f(x) = x is increasing on the entire real line; f(x) = -x is decreasing; f(x) = x^3 is increasing on all real numbers; f(x) = x^3 − 3x increases on (−∞, −1], decreases on [−1, 1], and increases again on [1, ∞). Sequences have parallel definitions, with increasing meaning a_{n+1} ≥ a_n and decreasing meaning a_{n+1} ≤ a_n.