Monotonocity

Let y = f (x) be a given function with 'D' as it's domain. Let ${\text{D}}_{\text{1}}\subseteq D$ then;

Increasing Function:

If a function f(x) is satisfying x₁ > x₂ $\Rightarrow$ f(x₁) > f (x₂) for all x_1, x₂$\in$ D₁, it means that the value of f (x) will keep on increasing with an increase in the value of x, then f is called increasing in D_1.

Decreasing Function:

If a function f(x) is satisfying x₁ > x₂ $\Rightarrow$ f(x₁) < f (x₂) for all x_1, x₂$\in$ D₁, it means that the value of f (x) will keep on increasing with an increase in the value of x, then f is called decreasing in D_1.

Non-Decreasing Function:

If a function f(x) is satisfying x₁ > x₂ $\Rightarrow$ f(x₁) $⩾$ f (x₂) for all x_1, x₂$\in$ D₁, it means that the value of f (x) will never decrease with an increase in the value of x, then f is called non-decreasing in D_1.

Non-Increasing Function:

If a function f(x) is satisfying x₁ > x₂ $\Rightarrow$ f(x₁) $⩽$f (x₂) for all x_1, x_2$\in$ D₁, it means that the value of f (x) will never increase with an increase in the value of x, then f is called non-increasing in D_1.

Note:

(i) If$f^{\prime }(x)\ge 0\forall x\in (a,b)$ and points which make $f^{\prime }(x)$equal to zero (in between (a, b)) don't form an interval, then f (x) would be increasing in [a, b] otherwise it will be non-decreasing function.

(ii) If$f^{\prime }(x)\le 0\forall x\in (a,b)$and points which make $f^{\prime }(x)$equal to zero (in between (a, b)) don't form an interval, f (x) would be decreasing in [a, b], otherwise it will be non-increasing.

Monotonic Function:

A function which is either increasing or decreasing in its domain is called a monotonic function.

Illustration 1: Prove that function f(x) = 2 cos x + cot x + 3x is decreasing in (0,$\pi$)

Solution: f'(x) = – 2 sin x – cosec ² x + 3

= - $\left[\frac{2{\sin }^{3}x+1-3{\sin }^{2}x}{{\sin }^{2}x}\right]=-\left[\frac{{(\sin x-1)}^2(2\sin x+1)}{{\sin }^{2}x}\right]$

2sin x + 1 > 0 in (0, $\pi$) hence f' > 0 function is increasing.