Monotonicity and Extremum of Functions

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What is Monotonicity and Extremum


Monotonicity: 

The nature of a function is governed by the sign of its derivative. If the graph of a function is in the upward direction or in the downward direction, then the function is named as a monotonic function. If the graph of the function is in the upward direction, then it has increasing values and it is said to be monotonically increasing. Similarly, if the graph of function is in the downward direction, then it has decreasing values and it is said to be monotonically decreasing.


Extremum: 

Points of the domain of a function where its graph changes its direction from upwards to downwards or from downwards to upwards is known as extremum. At such points, the derivative of a function, if it exists is necessarily zero.


Monotonic Functions

A function f(x) defined in the domain D is said to be:


i) Monotonic Increasing:

A function f(x) is said to be a monotonic increasing function if x₁ < x₂ and f(x₁) ≤ f(x₂). The graph of a monotonic increasing function can be represented as:

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ii) Monotonic Decreasing:

A function is said to be a monotonic decreasing function if x₁ < x₂ and f(x₁) ≥ f(x₂). The graph of a monotonic decreasing function can be represented as follows:

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iii) Strictly Increasing:

A function f(x) is said to be a strictly increasing function in its domain if x₂ > x₁ and f(x₂) > f(x₁) or dy / dx > 0. The graph of a strictly increasing function can be represented as follows:

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iv) Strictly Decreasing:

A function f(x) is said to be a strictly decreasing function in its domain if x₂ > x₁ and f(x₂) < f(x₁) or dy/dx < 0. The graph of a strictly decreasing function can be represented as follows:

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Method of Testing Monotonicity


1. Monotonic at a Point:

A function f(x) is said to be monotonic increasing or decreasing at a point x = a of its domain if it is monotonic increasing or decreasing in the interval (a - h, a + h), where h is a small positive number. Hence, it can be observed that if f(x) is monotonic increasing at x = a, then at this point tangent to its graph will make an acute angle with x-axis whereas if the function is monotonic decreasing, then tangent will make an obtuse angle with x-axis. Consequently f’(a) will be positive or negative according as f(x) is monotonic increasing or decreasing at x = a. So at x = a, function f(x) is monotonic increasing if f’(a) > 0 and monotonic decreasing if f’(a) < 0.


2. Monotonic in an Interval:

In [a, b], f(x) is – 

Monotonic increasing if f’(x) ≥ 0

Monotonic decreasing if f’(x) ≤ 0

Constant if f’(x) = 0 

Increasing if f’(x) > 0

Decreasing if f’(x) < 0

In above results, f’(x) should not be zero for all values of x, else f(x) will be a constant function. 

If in [a, b], f’(x) < 0 for at least for one value of x and f’(x) > 0 for at least for one value of x, then f(x) will not be monotonic in [a, b].


Maxima and Minima


Maxima

A function f(x) is said to attain a maximum at x = a if there exists a neighbourhood (a - δ , a + δ) such that

f(x) < f(a) for all x ∈ (a - δ, a + δ), x ≠ a

⇒ f(x) - f(a) < 0 for all x ∈ (a - δ, a + δ), x ≠ a

In such a case, f(x) is said to be the maximum value of f(x) at x = a.


Minima

A function f(x) is said to attain a minimum at x = a if there exists a neighbourhood (a - δ, a + δ) such that f(x) > f(a) for all x ∈ (a - δ, a + δ), x ≠ a

⇒ f(x) - f(a) > 0 for all x ∈ (a - δ, a + δ), x ≠ a

In such a case, f(x) is said to be the minimum value of f(x) at x = a.


Method of Testing for Maxima and Minima

For determining extreme values of a function f(x)

Step 1: Put y = f(x) 

Step 2: Find dy/dx 

Step 3: Put dy/dx = 0 and solve for x.

Step 4: To check the maxima or minima at x = a, first find the sign of f’(x) for values of x slightly less than a or greater than a.

  1. If  the sign of f’(x) changes from positive to negative, then f(x) has maximum at x = a.

  2. If  the sign of f’(x) changes from negative to positive, then f(x) has minimum at x = a.

  3. If  the sign of f’(x) does not change, then f(x) has neither maximum nor minimum at x = a.


Solved Examples


1. If f(x) = kx³ - 9x² + 9x + 3 s Monotonically Increasing in Each Interval then Find the Value of k.


Solution:

f’(x) = 3kx² - 18x + 9

f’(x) = 3(kx² - 6x + 3) > 0 ∀ x ∈ R


Therefore,

Δ = b² - 4ac < 0, k > 0

36 - 12k < 0

k > 3 


2. The Maximum Value of x³ - 12x² + 36x + 17 in the Interval [1, 10] is?


Solution:

Let f(x) = x³ - 12x² + 36x + 17.

Therefore,

f’(x) = 3x² - 24x + 36 = 0 at x = 2, 6

Again,

f’’(x) = 6x - 24 is negative at x = 2

So, f(6) = 17, f(2) = 49

At the end points, f(1) = 42, f(10) = 177.

Therefore, the maximum value of f(x) is 177.


Did You Know

  • If f(x) and g(x) are increasing or decreasing in [a, b] and gof is defined in [a, b], then gof is increasing.

  • If f(x) and g(x) are two monotonic functions in [a, b] such that one is increasing and the other is decreasing, then gof, if it is defined, is decreasing function.

  • If f(x) is strictly increasing in some interval, then in that interval, f⁻¹ exists ad that is also a strictly increasing function.

  • Between two equal values of f(x), there lies at least one maxima or minima.

  • Maxima and minima occur alternatively.

  • If f’(x) has the same sign on both sides of a point, then this function is neither maxima or minima.

  • When x passes a maximum point, the sign of f’(x) changes from positive to negative, whereas when x passes a minimum point, the sign of f’(x) changes from negative to positive.

  • If f(x) is maximum or minimum at x = a, then 1/f(x) [ f(x) ≠ 0 ] will be maximum or minimum at that point. 

FAQ (Frequently Asked Questions)

1. What are Some Standard Geometrical Results Related to Maxima and Minima?

The following results can easily be established:

  1. The area of a rectangle with a given perimeter is greatest when it is a square.

  2. The perimeter of a rectangle with a given area is least when it is a square.

  3. The greatest rectangle inscribed within a circle is a square.

  4. The greatest triangle inscribed within a circle is equilateral.

2. What is the Greatest and Least Value of a Function in a Given Interval?

If a function f(x) is defined in an interval [a, b], then the greatest or least values of this function occurs either at x = a or x = b or at those values of x where f’(x) = 0. The maximum value of a function f(x) in any interval [a, b] is not necessarily its greatest value in that interval.