What is a monotonically increasing function?

The correct option is: x₁ < x₂ ⇒ f(x₁) ≤ f(x₂) ∀ x₁, x₂ ∈ (a,b).

Understanding Monotonically Increasing Functions:

A function is defined as monotonically increasing (or non-decreasing) over a specific interval if the value of the function does not decrease as the input variable increases. Visually, as you trace the function from left to right along the horizontal axis, the graph will either rise or remain flat, but it will never curve downward.

Let's break down the mathematical statement step-by-step:

1. Selecting the Inputs: Let us choose any two arbitrary points, denoted as:

$x_1$

and

$x_2$

within a given open interval:

$(a,b)$

such that the first point lies to the left of the second point:

$x_1<x_2$

2. Comparing the Outputs: For the function to be monotonically increasing, the function's value at the larger input must be greater than or equal to its value at the smaller input. This is represented as:

$f\left(x_1\right)\le f\left(x_2\right)$

3. Generality (For All Points): This behavior must hold true for every possible pair of points in the interval. The symbol:

$\forall$

means "for all", and the symbol:

$\in$

means "in" or "belongs to". Therefore, the condition:

$\forall x_1,x_2\in (a,b)$

signifies that this relation applies universally across the entire interval.

Combining these pieces gives us the formal definition of a monotonically increasing function:

$x_1<x_2\Rightarrow f\left(x_1\right)\le f\left(x_2\right)\forall x_1,x_2\in (a,b)$