# Functions, Increasing and Decreasing

## Functions, Increasing and Decreasing

A function y *= f(x)* is called increasing over an interval *[a, b]* if for any pair of points *x* and*’, *a ≤ x ≤* the inequality *f(x)**≤f(x’)* is satisfied and strictly increasing if the inequality *f(x)< f(x’)* is satisfied. Decreasing and strictly decreasing functions are similarly defined. For example, the function *y = x2* (Figure 1, a) strictly increases over the interval [0, 1], and *y = l/(x* 1) (Figure 1, b) strictly decreases over this interval. Increasing functions are designated *as f(x)1* and decreasing functions *a.sf(x) I.* In order for a differentiable function *f(x)* to be increasing over an interval *[a, b],* it is necessary and sufficient that its derivative f’(x) be nonnegative over *[a, b].*

In addition to the increase and decrease of a function over an interval, the increase and decrease of a function at a point are considered. The function *y =f(x)* is called increasing at a point Jt0 if an interval (α, β) can be found containing point x_{0} such that for any point *x* from (α, β), *x >* x_{0}, the inequality *f(x _{0}) f(x)* is satisfied and that for any point jc from (α, β),

*x ≤ x*, the inequality

_{0}*f(x) f(x) ≤ f(x*is satisfied. The strict increase of a function at point x

_{0})_{0}is similarly defined.

*If f(x0) >*0, then the function/fjcj strictly increases at point

*x0. lff(x)*increases at every point of the interval

*(a, b),*then it increases over this interval.

### REFERENCE

Fikhtengol’ts, G. M.*Kurs differentsial’nogo i integral’nogo ischisleniia,*6th ed., vol. 1. Moscow, 1966.

S. B. STECHKIN