Monotonicity and Extremum of Functions

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) = ex are the examples of increasing function and f(x) = -x5 and f(x) = e-x are the examples of decreasing function.

Increasing function:

Increasing Function

If x1 < x2 and F(x1) < F(x2) then function is known as increasing function or strictly increasing function.

 

Decreasing function:

For F(x) = e(-x)

Decreasing Function

If x1 < x2 and F(x1) > F(x2) 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]

Non monotonic function

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

Monotonicity of a Function at a Point

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

Monotonicity of a function

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.

Monotonicity strickly 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) = x3 which is increasing at x = 0 although dy/dx = 0. This is because f(0 + h) > f(0) and f(0 – h) < f(0).

Monotonicity of Function Example

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.

Extremum of functions

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.

Extreme of Function

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)

The Extremum of a function

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

Interval of Monotonicity

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) = x3/3 + (5x)2/2 + 6x.

Solution: f(x) = x3/3 + (5x)2/2 + 6x

=> f’(x) = x+ 5x + 6

For strictly increasing:

f’(x) > 0

=> x2 + 5x + 6 > 0

=> x ϵ (∞,-3) U (-2,∞)

For strictly decreasing :

f’(x) < 0

=> x2 + 5x + 6 < 0

=> x ϵ (-3,-2).

Example 4: Find the extremum of function f(x) = 3x3 – 9x in the interval [-1, 4]

Solution: f(x) = 3x3 – 9x

=> f’(x) = 9x2 – 9 = 9(x2 – 1)

=> f’(x) = 0 => 9(x– 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 sinxcosx\sin x-\cos x increasing?

Solution:

We have, f(x)=cosx+sinxf'(x)=\cos x+\sin x

Now, f(x) is increasing function of x, if f(x)=cosx+sinx>0f'(x)=\cos x+\sin x>0 or

2cos(xπ4)>0\sqrt{2}\cos \left( x-\frac{\pi }{4} \right)>0 0x<3π4 i.e.f(x)>00\le x<\frac{3\pi }{4}\ i.e.\,\,\,f'(x)>0 in [0,3π4)\left[ 0,\frac{3\pi }{4} \right)

 

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