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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. \(\left\{\begin{array}{l}x_{1}<x_{2} \Rightarrow f\left(x_{1}\right) \leq f\left(x_{2}\right) \\\text { or } x_{1}<x_{2} \Rightarrow f\left(x_{1}\right)≯f\left(x_{2}\right) \end{array}\right.\), ∀x1, x2 ∈ D
or \(\left\{\begin{array}{l}x_{1}>x_{2} \Rightarrow f\left(x_{1}\right) \geq f\left(x_{2}\right) \\\text { or } x_{1}>x_{2}\Rightarrow f\left(x_{1}\right)≮ f\left(x_{2}\right)\end{array}\right.\), ∀x1, x2 ∈ 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. \(\left\{\begin{array}{l}x_{1}<x_{2} \Rightarrow f\left(x_{1}\right) \geq f\left(x_{2}\right) \\\text { or } x_{1}<x_{2} \Rightarrow f\left(x_{1}\right) ≮ f f\left(x_{2}\right)
\end{array}\right.\), ∀x1, x2 ∈ D
or \(\left\{\begin{array}{l}x_{1}>x_{2} \Rightarrow f\left(x_{1}\right) \leq f\left(x_{2}\right) \\\text { or } x_{1}>x_{2} \Rightarrow f\left(x_{1}\right) ≯f f\left(x_{2}\right)\end{array}\right.\), ∀x1, x2 ∈ 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 x1 < x2 ⇒ f(x1) < f(x2) ∀ x1, x2 ∈ D, then f(x) is called strictly increasing in domain D.
(v) If x1 < x2 ⇒ f(x1) > f(x2), ∀ x1, x2 ∈ 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”.
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