What is the mathematical expression for monotonically decreasing function?

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

Step-by-Step Explanation:

1. Understanding Monotonicity:
A function is described as monotonic if it preserves the order of elements from its domain to its codomain. In simpler terms, it either consistently increases or consistently decreases as the input values increase.

2. Defining a Decreasing Trend:
For a function to be decreasing, larger input values must yield smaller (or equal) output values. Let us take any two arbitrary points, $x_1$ and $x_2$, in the interval $(a,b)$ such that $x_1$ is strictly less than $x_2$:

$x_1<x_2$

3. Applying the Function:
If the function $f$ is monotonically decreasing, the outputs must reverse this order. This means the value of the function at the smaller input, $f\left(x_1\right)$, must be greater than or equal to the value of the function at the larger input, $f\left(x_2\right)$:

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

4. Universal Applicability:
This condition must hold true for all possible pairs of elements within the given interval $(a,b)$. Mathematically, this is expressed using the universal quantifier $\forall$ (which means "for all"):

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

5. Combining into the Final Expression:
Putting these steps together gives the formal mathematical definition of a monotonically decreasing function on an interval $(a,b)$:

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