Functions are said to be monotonic if they are either increasing or decreasing in their entire domain
e.g. `f(x) = e^x ; f(x) = ln x` & `f(x) = 2x + 3` are some of the examples of functions which are increasing
where as `f(x) =- x^3 ; f(x) e^(-x)` and `f(x) = cot^(-1) (x)` are some of the examples of the functions which
are decreasing.


Increasing function :
If `x_1 < x_2` and `f(x _1) < f(x _2)` then function is called increasing function or strictly increasing function.


Decreasing function :

`f(x) = e^(-x)`
If `x_1 < x_2 `
but `f(x _1)> f(x_2)` in entire domain then fucntion is called
decreasing function or strictly decreasing function.

Functions which are increasing as well as decreasing in their domain are said to be non monotonic

e.g. `f(x) =sin x ; f(x) =ax^2 +bx+c` and `f(x) = |x|`, however in the interval `[0,pi/2 ]`, `f(x) =sin x` will be said to be increasing.

A function is said to be monotonically increasing `x = a` if `f(x)`
satisfies

`tt( (f (a+h) > f (a)) ,( f (a-h) < f(a))]_(for a small positive h.)

A function is said to be monotonically decreasingat `x =a` if `f(x)`
satisfies

`tt( (f (a+h) < f (a)) ,( f (a-h) > f(a))]_(for a small positive h.)


Note : It should be noted that we can talk of monotonocity of `f(x)` at `x = a` only if `x = a` lies in the domain of `f(x)`,
without any consideration of continuity or differentiability of `f(x)` at `x = a`.

(a) For an increasing function in some interval,
if `Delta x > 0 <=> Delya y > 0` or `Delta x < 0 <=> Delta y < 0`
then `f` is said to be monotonic (strictly) increasing in that interval. In
other words if `Delta y` and `Delta X` have the same sign i.e. `dy/dx > 0` ,
for increasing function . Hence if `dy/dx > 0` in some (interval) then `y` yis said to be increasing
function in that interval and conversely if `f(x)` is increasing in some mterval then
`dy/dx > 0` in that `J`


(b) Sumlarly If `dy/dx < 0` in some interval then `y` is decreasing in that `J` and conversely.

Note: Hence to find the intervals of monotonocity for a function `y =f(x)` one has to find `dy/dx` and solve the
inequality , `dy/dx > 0` or `dy/dx<0` . The solution of this inequality gives the interval of monotonocrty.

(a) `dy/dx` at some point may be equal to zero but `f(x)` may still be increasing
at `x = a`. Consider `f (x) = x^3` which is increasing at `x = 0` although
`f' (x)=0`. This is because `f(0 + h) > f(0)` and `f(0 - h) < f(0)`. At all
such points where `dy/dx =0` but `y` is still increasing or decreasing are
known as point of inflection, which indicate the change of concavity
of the curve.

(b) If `f` is increasing for `x > a` and `f` is also increasing for `x < a` then
`f` is also increasing as `x = a` provided `f(x)` is continuous at `x = a`.

(c) If `f(x)` is discontinuous at `x = a` then one can draw the graph as shown
`x =a` is the point of maxima

(i) A function `f(x)` is said to be monotonically increasing for all such interval `(a, b)` where `f '(x) ge 0` and
equality may hold only for discrete values of `x`, i. e. , `f'(x)` does not identically become zero for `x in (a, b)`
or any sub-interval.

(ii) `f(x)` is said to be monotonically decreasing for a ll such interval `(a, b)` where `f'(x) le 0` and equility may
hold only for discrete values of `x`.

Note : By discrete points, we mean those points where `f'(x) = 0` does not form any interval.

`f(x)` is said to be non-decreasing in domain if for every
`x _1, x_2 in D_1, x_1 > x_2 => f(x _1) ge f(x_ 2)`. It means that the
value of `f( x)` would never decrease with an increase in
the value of `x` (Figure).

Non Increasing function :

`f(x)` is said to be non-increasing in domain if for every
`x_1, x_2 in D_1, x_1 > x_2 f(x _1) le f(x _2)`. It means that the
value of `f(x)` would never increase with an increase in
the value of `x` (Figure).

Let us consider another function whose graph is shown
for `x in (a, b)`.

Here also `f '(x) ge 0`, for all `x in (a, b)` but note that in this
case equality of `f'(x) = 0` holds for all `x in (c, d)` and
`(e, b)`. Here `f'(x)` becomes identically zero and hence
the given function cannot be assumed to be
monotonically increasing for `x in (a, b)`.

Note:
(i) If a function is monotonic at `x = a` it can not have extremum point at `x = a` and vice versa i. e. a point on
the curve can not simultaneously be an extremum as well as monotonic point.

(ii) If `f` is increasing then nothing definite can be said about the function `f '(x)` w.r.t. its increasing or decreasing
behaviour.