Monotonicity is one of the important concepts of application of derivatives. The monotonicity of a function gives an idea about the behaviour of the function. A function is said to be monotonically increasing if its graph is only increasing with increasing values of equation. Similarly, function is monotonically decreasing if its values are only decreasing.

**Monotonicity: **The most useful element taken into consideration amongst the total activities of the function is their monotonic behaviour. It tells about increasing or decreasing behaviour of the function.

**Extremum: **An extremum of a function is the point where we get the maximum or minimum value of the function in some interval.

## Monotonicity of a Function

**What is monotonic function?** Functions are known as monotonic if they are increasing or decreasing in their entire domain.

Examples : f(x) = 2x + 3, f(x) = log(x), f(x) = e^{x} are the examples of increasing function and f(x) = -x^{5 }and f(x) = e^{-x} are the examples of decreasing function.

**Increasing function:**

If x_{1} < x_{2} and F(x_{1}) < F(x_{2}) then function is known as increasing function or strictly increasing function.

**Decreasing function:**

For F(x) = e^{(-x)}

If x_{1} < x_{2} and F(x_{1}) > F(x_{2}) then function is known as decreasing function or strictly decreasing function.

### Non monotonic Function

The functions which are increasing as well as decreasing in their domain are known as non monotonic function.

**Example: **f(x) = sin x , f(x) = |x| are examples of non monotonic function. But f(x) = sin x is increasing in [0, Π/2] or we can say it is monotonic in [0, Π/2]

## Monotonicity of a function at a point

A function is said to be monotonically increasing at x = a if it satisfies

f(a + h) > f(a)

f(a – h) < f(a)

Where h is very small value

A function is said to be monotonically decreasing at x = a if it satisfies

f(a + h) < f(a)

f(a – h) > f(a)

Where h is very small value

Note: We can talk about the monotonicity of a function at x = a only if x = a is in the domain of the function and we don’t need to take continuity and differentiability in consideration.

## Monotonicity in an Interval

(a). For an increasing function in some interval,

If dy/dx > 0 for all the values of x belongs to that interval , then function is known as monotonically increasing or strictly increasing function.

(b). For an decreasing function in some interval,

If dy/dx < 0 for all the values of x belongs to that interval , then function is known as monotonically decreasing or strictly decreasing function.

Note: hence to find the interval of monotonicity for a function y = f(x) we need to find out the value of dy/dx and have to solve the inequality dy/dx > 0 0r dy/dx < 0. The solution of this inequality gives the interval of monotonicity.

Note: If dy/dx = 0 for a function y = f(x), still the function can be increasing at x = a. Consider a function f(x) = x^{3} which is increasing at x = 0 although dy/dx = 0. This is because f(0 + h) > f(0) and f(0 – h) < f(0).

All those points where dy/dx = 0 but function is still increasing or decreasing are known as Point of Inflection, which indicate the change of concavity.

## Increasing and Decreasing Function

(a). For a function y = f(x) to be monotonically increasing dy/dx ≥ 0 for all such values of interval (a,b) and equality may hold for discrete values.

(b). For a function y = f(x) to be monotonically decreasing dy/dx ≤ 0 for all such values of interval (a,b) and equality may hold for discrete values.

Note: by discrete points, we mean all those points where dy/dx = 0 does not form any interval.

## Maxima and Minima

- Local Maxima: is defined as the point on the curve where the function value at the point is greater than the limiting function value.
- Local Minima: is defined as the point on the curve where the function value at the point is lesser than the limiting function value.
- Global Maxima: is the highest value of the function among the various critical points of the function.
- Global Minima: is the least value of the function among the various critical points of the function.

## Extremum of functions

Extremum of functions is the least and the greatest values of the function. There are three different cases for all such values and those are:

**Case 1 :** If a function y = f(x) is the strictly increasing function in an interval [a,b], then f(a) is the least value and f(b) is the greatest value as shown in figure-1.

**Case 2 : **If a function y = f(x) is the strictly decreasing function in an interval [a,b], then f(a) is the greatest value and f(b) is the lowest value as shown in figure-2.

**Case 3 :** if a function y = f(x) is non monotonic in interval [a,b] and is continuous then the greatest and the least value of the function are at those points where dy/dx = 0 or where dy/dx does not exist or at extreme values i.e. at x = a and x = b.(figure-3)

## Points to remember

1] A function might consist of many local maxima and local minima but only one global maximum and one global minimum.

2] The value of the local maximum or local minimum may or may not be the global maxima or global minima.

3] The value of the local maximum may be less than a local minimum at some point.

4] For any given continuous function, the happening of minima and maxima is alternate.

## Examples on Monotonicity and Extremum of functions

**Example 1:** Prove that f(x) = x – sin(x) is an increasing function.

**Solution:** f(x) = x – sin(x)

=> dy/dx = 1 – cos(x)

dy/dx ≥ 0 as cos(x) having value in interval [-1,1] and dy/dx = 0 for the discrete values of x and do not form an interval, hence we can include this function in monotonically increasing function.

**Example 2: **Find the interval of monotonicity for f(x) = x/(log(x)).

**Solution: **f(x) = x/(log(x))

=> dy/dx = (log(x) – 1)/(log(x))^2

Sign of dy/dx

So we can say that function is increasing in (e,∞) and decreasing in (0,1) U (1,e).

**Example 3: **Find the interval of monotonicity for f(x) = f(x) = x^{3}/3 + (5x)^{2}/2 + 6x.

**Solution: **f(x) = x^{3}/3 + (5x)^{2}/2 + 6x

=> f’(x) = x^{2 }+ 5x + 6

For strictly increasing:

f’(x) > 0

=> x^{2} + 5x + 6 > 0

=> x ϵ (∞,-3) U (-2,∞)

For strictly decreasing :

f’(x) < 0

=> x^{2} + 5x + 6 < 0

=> x ϵ (-3,-2).

**Example 4:** Find the extremum of function f(x) = 3x^{3 }– 9x in the interval [-1, 4]

**Solution: **f(x) = 3x^{3} – 9x

=> f’(x) = 9x^{2} – 9 = 9(x^{2} – 1)

=> f’(x) = 0 => 9(x^{2 }– 1) = 0 or x = ±1

=> f(-1) = 6

=> f(1) = – 6

=> f(4) = 156

=> Greatest value = 156 and least value = – 6.

**Example 5:** In what interval is the function

**Solution:**

We have,

Now, f(x) is increasing function of x, if