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Increasing and Decreasing Functions

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Let I be an interval contained in the domain of a real valued function f. Then f is said to be 
(i) increasing on I if `x_1` < `x_2` in I ⇒ `f(x_1)` < `f(x_2)` for all `x_1`, `x_2` ∈ I. 
(ii) decreasing on I, if `x_1`,` x_2` in I ⇒ `f(x_1)` < `f(x_2)` for all `x_1, x_2` ∈ I. 
(iii) constant on I, if f(x) = c for all x ∈ I, where c is a constant.
(iv) decreasing on I if `x_1` < `x_2` in I ⇒ `f(x_1)` ≥ `f(x_2)` for all `x_1`,` x_2` ∈ I. 
(v) strictly decreasing on I if `x_1` < `x_2` in I ⇒ `f(x_1)` > `f(x_2)` for all ` x_1`, `x_2` ∈ I.
For graphical representation of such functions see Fig.


Let `x_0` be a point in the domain of definition of a real valued function  f. Then  f  is said to be increasing, decreasing at `x_0` if  there exists an open interval I containing `x_0` such that f is increasing, decreasing, respectively, in I.


Let  f  be continuous on [a, b] and differentiable on the open interval (a,b). Then 
(a) f  is increasing in [a,b] if f′(x) > 0 for each x ∈  (a, b) 
(b) f  is decreasing in [a,b] if f′(x) < 0 for each x ∈ (a, b) 
(c) f  is a constant function in [a,b] if f′(x) = 0 for each x ∈ (a, b)

Proof:  (a) Let `x_1, x_2` ∈ [a, b] be such that `x_1 < x_2`.
Then, by Mean Value Theorem (Theorem 8 in Chapter 5), there exists a point c between `x_1` and `x_2` such that 
`f(x_2) – f(x_1) = f′(c) (x_2 – x_1)` 
i.e. `f(x_2) – f(x_1) > 0`        (as f′(c) > 0 (given))

i.e.` f(x_2) > f(x_1)`

Thus, we have `x_1< x_2` `f(x_1)` `f(x_2)` ,  for all `x_1,x_2` [a,b]

Hence, f  is an increasing function in [a,b].
The proofs of part (b) and (c) are similar. It is left as an exercise to the reader.

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