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2. MONOTONICITY OF A FUNCTION

In this section we concentrate on some applications of the derivatives. Firstly we have to introduce some important notions.

Definition

Let f be a function defined on an interval P. We say that the function f is:

  1. nondecreasing on P, if for any x1, x2 ∈ P the condition x1 < x2 implies f (x1) ≤ f (x2),

  2. increasing on P, if for any x1, x2 ∈ P the condition x1 < x2 implies f (x1) < f (x2).

In a similar way one defines a nonincreasing function and a decreasing function.

If a function f is either nondecreasing or increasing or nonincreasing or decreasing, we say that the function f is monotone.

We will apply the derivatives for investigation of the monotonicity of a function.

Remark

Let f be differentiable at x0.

  1. If the function f is nondecreasing on some neighborhood of x0, then f ' (x0) ≥ 0.

  2. If the function f is nonincreasing on some neighborhood of x0, then f ' (x0) ≤ 0.

The following theorem states the sufficient conditions for a function to be monotone.

Theorem

Let f be a function continuous on an interval P and differentiable in its interior.

  1. If f ' (x) > 0 for every x ∈ Int P, then f is increasing on P;
  2. If f ' (x) ≥ 0 for every x ∈ Int P, then f is nondecreasing on P;
  3. If f ' (x) < 0 for every x ∈ Int P, then f is decreasing on P;
  4. If f ' (x) ≤ 0 for every x ∈ Int P, then f is nonincreasing on P.

Proof. We will prove point (1). Proofs of remaining points are similar. Let a, b ∈ P, a < b. We will apply the mean value theorem for the interval [a, b]. We have:

for some c ∈ (a, b) ⊂ P. By the assumption f ' (c) > 0 and b - a > 0, therefore f (b) - f (a) > 0.

Remark

The condition (f ' (x) > 0 for every x ∈ Int P) is not a necessary condition, for the function f to be increasing on P.

Example

Consider the function f (x) = x³. It is known that it is an increasing function on R, but f ' (x) = 3x², that is f ' (x) ≥ 0.

The necessary and sufficient condition for a function f to be increasing on an interval P is delivered by the following theorem:

Theorem

If a function f is continuous in an interval P, differentiable in the interior of P, then f is increasing in the interval P iff, the following conditions are satisfied

  1. f ' (x) ≥ 0 for every x ∈ Int P
  2. function f ' is not identically equal to 0 in any non-empty open interval contained in P.

Example

The function f (x) = x-sinx is increasing in the interval P = (-∞, ∞).

Indeed, it is continuous in the interval P and f ' (x) = 1 - cosx ≥ 0 for every x∈ IntP = P, and the function f ' is not identically equal to 0 in any non-empty open interval contained in P.


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