Monotonicity

Have you ever wondered how Mathematicians predict the behavior of functions without graphing every single point? The answer lies in the concept of monotonicity. Imagine you're hiking up a mountain. The path could steadily climb upwards, making each step a bit higher than the last, or it could take you downhill, each step lower than the previous one. In mathematics, we describe functions in a similar way: those that consistently increase are called "increasing functions," while those that consistently decrease are "decreasing functions." Together, these are known as monotonic functions.

Monotonicity is a fundamental idea that helps us understand and analyze how functions behave. It's like having a map that tells you whether the path ahead will keep climbing or start descending. This concept is not just theoretical—it has practical applications in fields ranging from economics to engineering, where predicting trends and behaviors is crucial.

In this article, we will delve into the world of monotonicity. We'll explore what it means for a function to be monotonic, how to identify such functions, and why this concept is so important. Whether you're a student, a teacher, or just a math enthusiast, you'll find this journey both enlightening and enjoyable. So, let's get started and unlock the secrets of monotonicity together!

1.0Monotonic Function Definition

A monotonic function is a function that preserves a consistent order in its output values as its input values change. 

1. Monotonically Increasing Function:

• A function f(x) is said to be monotonically increasing if for every pair of input values x₁ and x₂ such that x₁ < x₂, the corresponding output values satisfy f(x₁) ≤ f(x₂). 
• This means that as x increases, f(x) does not decrease; it either increases or stays the same.

2. Strictly Increasing Function:

• A function f(x) is strictly increasing if for every pair of input values x₁ and x₂ such that x₁ < x₂, the corresponding output values satisfy (x₁) < f(x₂). 
• This means that as x increases, f(x) always increases.

3. Monotonically Decreasing Function:

• A function g(x) is called monotonically decreasing if for every pair of input values x₁ and x₂ such that x₁ < x₂, the corresponding function values satisfy g(x₁) ≥ g(x₂). 
• This means that as x increases, g(x) does not increase; it either decreases or stays the same.

4. Strictly Decreasing Function:

• A function g(x) is strictly decreasing if for every pair of input values x₁ and x₂ such that x₁ < x₂, the corresponding function values satisfy g(x₁) > g(x₂). 
• This means that as x increases, g(x) always decreases.

In summary, a monotonic function is one that maintains a consistent trend, either non-decreasing or non-increasing, across its domain. This characteristic makes it easier to predict and analyze the function's behavior without needing to graph it extensively. Monotonic functions play a crucial role in various fields of mathematics, including calculus, analysis, and applied disciplines such as economics and engineering.

2.0Why is the Monotonicity Principle Important?

The monotonicity principle is crucial because it allows mathematicians and analysts to make significant inferences about the behavior of functions:

• Simplifies Analysis: Understanding whether a function is increasing or decreasing helps simplify the analysis of its behavior, making it easier to predict and interpret its graph and trends.
• Optimization Problems: In optimization, monotonic functions can indicate whether a function has a maximum or minimum value over a given interval, aiding in the identification of optimal solutions.
• Economic and Scientific Applications: Monotonicity is widely used in economics to model growth trends and in science to describe natural phenomena, such as population growth or decay rates.

By understanding and applying the monotonicity principle, one can gain deeper understanding into the properties of functions and their real-world applications, making it an essential tool in the mathematical toolkit.

3.0Monotonicity of a Function in the Neighbourhood of a Point

1. First Derivative Test:

• If f'(x) > 0 for all x in some interval (a − δ, a + δ) where δ is a small positive number, then f(x) is increasing at x = a.
• If f'(x) < 0 for all x in some interval (a − δ, a + δ), then f(x) is decreasing at x = a.

2. Second Derivative Test for Concavity (which influences monotonicity):

• If f''(x) > 0 at x = a, the function is concave up in the neighborhood of a, suggesting f'(x) is increasing around a.
• If f''(x) < 0 at x = a, the function is concave down in the neighborhood of a, suggesting f'(x) is decreasing around a.

4.0Monotonicity in an Interval

For a function f(x) defined in an interval, its monotonic behavior can be described as follows:

1. Increasing Function:

• If Δx > 0 implies Δy > 0 or Δx < 0 implies Δy < 0 within the interval, then the function f(x) is said to be strictly increasing in that interval.
• Equivalently, for a differentiable function, if $\frac{\mathrm{dy}}{\mathrm{dx}}>0$ in an interval, then y is an increasing function in that interval. Conversely, if the function f(x) is increasing in some interval, then its derivative $\frac{\mathrm{dy}}{\mathrm{dx}}>0$ within that interval.

2. Decreasing Function:

• If Δx > 0 implies Δy < 0 or Δx < 0 implies Δy > 0 within the interval, then the function f(x) is said to be strictly decreasing in that interval.
• Similarly, for a differentiable function if $\frac{dy}{dx}<0$ in an interval, then y is a decreasing function in that interval. Conversely, if the function f(x) is decreasing in some interval, then its derivative $\frac{dy}{dx}<0$ within that interval.

In summary, for a differentiable function f(x) to be classified as strictly increasing in an interval, its derivative $\frac{dy}{dx}$ must be positive throughout that interval. Similarly, for the function to be classified as strictly decreasing, its derivative $\frac{dy}{dx}$ must be negative throughout the interval.

Note: To determine the interval of monotonicity for a function y = f(x), follow these steps:

1. Calculate the derivative $\frac{dy}{dx}.$

2. Solve the inequality $\frac{dy}{dx}>0.$

The solution to this inequality provides the interval where the function f(x) is increasing. Similarly, solving $\frac{\mathrm{dy}}{\mathrm{dx}}<0$, will give the interval where the function is decreasing.

Monotonicity of Differentiable Functions

Consider an interval $I\left(\subseteq D_f\right)$

that can be [a, b] or (a, b) or [a, b) or (a, b]. 

(1) f' (x) > 0 ∀ x ∈ I ⇒ f   is strictly increasing function over the interval I. 

(2) f' (x) ≥ 0 ∀ x ∈ I ⇒ f is increasing function over the interval I. 

(3) f' (x) > 0 ∀ x ∈ I and f'(x) = 0 do not form any interval (that means f'(x)=0 at discrete points)

⇒ f is strictly increasing function over the interval I.

(4) f' (x) < 0 ∀ x ∈ I ⇒ f is strictly decreasing function over the interval I.

(5) f' (x) ≤ 0 ∀ x ∈ I ⇒ f is decreasing function over the interval I.

(6) f' (x) ≤ 0 ∀ x ∈ I and f'(x) = 0 do not form any interval (that means f'(x)=0 at discrete points)

⇒ f is strictly decreasing function over the interval I.

5.0Graphs Showing Increasing and Decreasing Functions

6.0Solved Example of Monotonicity

Example 1: Prove that the function f(x)= $\log \left(x^3+\sqrt{x^6+1}\right)$ is strictly increasing.

Solution: Now,  f(x)= $\log \left(x^3+\sqrt{x^6+1}\right)$

$f^{\prime }(x)=\frac{1}{x^3+\sqrt{x^6+1}}\left(3x^2+\frac{6x^5}{2\sqrt{x^6+1}}\right)=\frac{3x^2}{\sqrt{x^6+1}}\ge 0$

⇒ f(x) is strictly increasing.

Example 2: Which of the following functions are strictly decreasing on $\left(0,\frac{\pi }{2}\right)$

(A) cos x (B) cos 2x (C) cos 3x (D) tan x

Ans. (A, B)

Solution:

Non-monotonic

f(x) = tan x is strictly increasing in $\left(0,\frac{\pi }{2}\right)$

Option A and B are correct.

Example 3: The function f, defined by $f(x)=(x+2)e^{-x}$ is

(A) decreasing for all x                                            

(B) decreasing in $(-\infty ,-1)$ and increasing $(-1,\infty )$ in 

(C) increasing for all x                                            

(D) decreasing in $(-1,\infty )$ and increasing in $(-\infty ,-1)$

Solution:

f(x) = (x + 2) e^-x

f'(x) = (x+2) e^–x (–1) + e^–x = –e^–x (x+1)

Increasing if f'(x) > 0 ⇒  –e^–x (x+1) > 0 

⇒ (x+1) < 0 [∵ e^–x always positive]

x ∈ (–∞, –1) & decreasing in (–1, ∞) as f'(x) < 0.

Example 4: The function $f(x)=\cos x-2px$ is monotonically decreasing for   

(A) $p<\frac{1}{2}$ (B) $p>\frac{1}{2}$ (C) $\mathrm{p}<2$ (D) $\mathrm{p}>2$

Ans. (B)

Solution:

f(x) = cos x – 2px

f'(x) = –sin x – 2p

For monotonic decreasing f'(x) < 0

⇒  2p > –sin x

$\Rightarrow p>\frac{1}{2}$

   [∵ –1 ≤ sin x ≤ 1]

Example 5: If graph of f(x) = 3x⁴ + 2x³ + ax² – x + 2 is concave up for all real x, find values of a 

Solution:

f’’(x) = 36x² + 12x + 2a

f’’(x) > 0 ⇒ 36x² + 12x + 2a ≥ 0 ∀ x ∈ R

D ≤ 0

⇒ 12² – 4(36) (2a) ≤ 0

=> 1 – 2a ≤ 0

Therefore, a ≥ 1/2