# Monotonicity Formulas

Have you ever wondered how to solve Monotonicity Related Problems easily? You can try out solving the problems with Monotonicity Formulae provided. Apply the Monotonicity Formulas during your homework or assignments and arrive at the solution quickly. Master the concept of Monotonicity by referring to the Monotonicity Formulas Cheat Sheet & Tables.

## Monotonicity Formulas Cheat Sheet

Have a look at the Monotonicity Formulas listed below and try to memorize them so that you can solve your problems simply. Learn the Important Formulas on Monotonicity and do your calculations at a faster pace instead of lengthy steps. Get a good grip on the fundamentals of the Monotonicity Concept taking help from Monotonicity Formulae Sheet & Tables.

1. Monotonic Function

These are of two types

(i) Monotonic Increasing

If the value of f(x) should increase (decrease) or remain equal by increasing (decreasing) the value of x.

i.e. \(\left\{\begin{array}{l}x_{1}<x_{2} \Rightarrow f\left(x_{1}\right) \leq f\left(x_{2}\right) \\\text { or } x_{1}<x_{2} \Rightarrow f\left(x_{1}\right)≯f\left(x_{2}\right) \end{array}\right.\), ∀x_{1}, x_{2} ∈ D

or \(\left\{\begin{array}{l}x_{1}>x_{2} \Rightarrow f\left(x_{1}\right) \geq f\left(x_{2}\right) \\\text { or } x_{1}>x_{2}\Rightarrow f\left(x_{1}\right)≮ f\left(x_{2}\right)\end{array}\right.\), ∀x_{1}, x_{2} ∈ D

(ii) Monotonic Decreasing

If the value of f(x) should decrease (increase) or remain equal by increasing (decreasing) the value of x.

i.e. \(\left\{\begin{array}{l}x_{1}<x_{2} \Rightarrow f\left(x_{1}\right) \geq f\left(x_{2}\right) \\\text { or } x_{1}<x_{2} \Rightarrow f\left(x_{1}\right) ≮ f f\left(x_{2}\right)

\end{array}\right.\), ∀x_{1}, x_{2} ∈ D

or \(\left\{\begin{array}{l}x_{1}>x_{2} \Rightarrow f\left(x_{1}\right) \leq f\left(x_{2}\right) \\\text { or } x_{1}>x_{2} \Rightarrow f\left(x_{1}\right) ≯f f\left(x_{2}\right)\end{array}\right.\), ∀x_{1}, x_{2} ∈ D

(iii) A function is said to be monotonic function in a domain if it is either monotonic increasing or monotonic decreasing in that domain.

(iv) If x_{1} < x_{2} ⇒ f(x_{1}) < f(x_{2}) ∀ x_{1}, x_{2} ∈ D, then f(x) is called strictly increasing in domain D.

(v) If x_{1} < x_{2} ⇒ f(x_{1}) > f(x_{2}), ∀ x_{1}, x_{2} ∈ D then it is called strictly decreasing in domain D.

2. Method of testing monotonicity

(i) At a point x = a, function f(x) is

Monotonic increasing ⇒ f’ (a) > 0

Monotonic deacreasing ⇒ f’ (a) < 0

(ii) In an interval

A function f(x) defined in the intervel [a, b] will be

Monotonic increasing ⇒ f’(x) ≥ 0
Monotonic decreasing ⇒ f’(x) ≤ 0 Constant ⇒ f’(x) = 0 ∀ x ∈ (a, b) Strictly increasing ⇒ f’(x) > 0 Strictly decreasing ⇒ f’(x) < 0 |

Note: A strictly monotomic function is always one one and onto or “BIJECTIVE”.