monotonicity criterion

Suppose that f:[a,b] is a functionMathworldPlanetmath which is continuousMathworldPlanetmath on [a,b] and differentiableMathworldPlanetmathPlanetmath on (a,b).

Then the following relations hold.

  1. 1.

    f(x)0 for all x(a,b) f is an increasing function on [a,b];

  2. 2.

    f(x)0 for all x(a,b) f is a decreasing function on [a,b];

  3. 3.

    f(x)>0 for all x(a,b) f is a strictly increasing function on [a,b];

  4. 4.

    f(x)<0 for all x(a,b) f is a strictly decreasing function on [a,b].

Notice that the third and fourth statement cannot be inverted. As an example consider the function f:[-1,1], f(x)=x3. This is a strictly increasing function, but f(0)=0.

Title monotonicity criterion
Canonical name MonotonicityCriterion
Date of creation 2013-03-22 13:45:12
Last modified on 2013-03-22 13:45:12
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 7
Author paolini (1187)
Entry type Theorem
Classification msc 26A06