Maxima and Minima

A function f(x) is said to have a local maximum at x = a, if $ is a neighbourhood I of 'a', such that

f(a) f(x) for all x I

The number f(a) is called the local maximum of f(x). The point a is called the point of maximum.

Note that when 'a' is the point of local maxima, f(x) is increasing for all values of x < a and f (x) is decreasing for all values of x > a in the given interval.

At x = a, the function ceases to increase.

A function f(x) is said to have a local minimum at x = a, if $ is a neighbourhood I of 'a', such that

f(a) f(x) for all x I

Here, f(a) is called the local minimum of f(x). The point a is called the point of minimum.

Note that, when a is a point of local minimum f (x) is decreasing for all x < a and f (x) is increasing for all x > a in the given interval. At x = a, the function ceases to decrease.

If f(a) is either a maximum value or a minimum value of f in an interval I, then f is said to have an extreme value in I and the point a is called the extreme point.

Monotonic Function

A function is said to be monotonic if it is either increasing or decreasing but not both in a given interval.

Consider the function

The given function is increasing function on R. Therefore it is a monotonic function in [0,1]. It has its minimum value at x = 0 which is equal to f (0) =1, has a maximum value at x = 1, which is equal to f (1) = 4.

Here we state a more general result that, 'Every monotonic function assumes its maximum or minimum values at the end points of its domain of definition.'

Note that 'every continuous function on a closed interval has a maximum and a minimum value.'

Theorem 2:

(First Derivative Test)

Let f (x) be a real valued differentiable function. Let a be a point on an interval I such that f '(a) = 0.

(a) a is a local maxima of the function f (x) if

i) f (a) = 0

ii) f(x) changes sign from positive to negative as x increases through a.

That is, f (x) > 0 for x < a and

f (x) < 0 for x > a

(b) a is a point of local minima of the function f (x) if

i) f (a) = 0

ii) f(x) changes sign from negative to positive as x increases through a.

That is, f (x) < 0 for x < a

f (x) > 0 for x > a

Working Rule for Finding Extremum Values Using First Derivative Test

Let f (x) be the real valued differentiable function.

Step 1:

Find f '(x)

Step 2:

Solve f '(x) = 0 to get the critical values for f (x). Let these values be a, b, c. These are the points of maxima or minima.

Arrange these values in ascending order.

Step 3:

Check the sign of f'(x) in the immediate neighbourhood of each critical value.

Step 4:

Let us take the critical value x= a. Find the sign of f '(x) for values of x slightly less than a and for values slightly

greater than a.

(i) If the sign of f '(x) changes from positive to negative as x increases through a, then f (a) is a local maximum value.

(ii) If the sign of f '(x) changes from negative to positive as x increases through a, then f (a) is local minimum value.

(iii) If the sign of f (x) does not change as x increases through a, then f (a) is neither a local maximum value not a minimum value. In this case x = a is called a point of inflection.

Example:

Find the local maxima or local minima, if any, for the following function using first derivative test

f (x) = x³ - 6x² + 9x + 15

Solution:

f (x) = x³ - 6x² + 9x + 15

f ' (x) = 3x² -12x + 9

= 3(x²- 4x + 3)

= 3 (x - 1) (x - 3)

Thus x = 1 and x = 3 are the only points which could be the points of local maxima or local minima.

Let us examine for x=1

When x<1 (slightly less than 1)

f '(x) = 3 (x - 1) (x - 3)

= (+ ve) (- ve) (- ve)

= + ve

When x >1 (slightly greater than 1)

f '(x) = 3 (x -1) (x - 3)

= (+ ve) (+ ve) (- ve)

= - ve

The sign of f '(x) changes from +ve to -ve as x increases through 1.

x = 1 is a point of local maxima and

f (1) = 1³ - 6 (1)² + 9 (1) +15

= 1- 6 + 9 + 15 =19 is local maximum value.

Similarly, it can be examined that f '(x) changes its sign from negative to positive as x increases through the point x = 3.

\ x = 3 is a point of minima and the minimum value is

f (3) = (3)³- 6 (3)²+ 9(3) + 15

= 15

Theorem 3: (Second Derivative Test)

Let f be a differentiable function on an interval I and let a I. Let f "(a) be continuous at a. Then

i) 'a' is a point of local maxima if f '(a) = 0 and f "(a) < 0

ii) 'a' is a point of local minima if f '(a) = 0 and f "(a) > 0

iii) The test fails if f '(a) = 0 and f "(a) = 0. In this case we have to go back to the first derivative test to find whether 'a' is a point of maxima, minima or a point of inflexion.

Working Rule to Determine the Local Extremum Using Second Derivative Test

Step 1

For a differentiable function f (x), find f '(x). Equate it to zero. Solve the equation f '(x) = 0 to get the Critical values of f (x).

Step 2

For a particular Critical value x = a, find f "^'(a)

(i) If f ''(a) < 0 then f (x) has a local maxima at x = a and f (a) is the maximum value.

(ii) If f ''(a) > 0 then f (x) has a local minima at x = a and f (a) is the minimum value.

(iii) If f ''(a) = 0 or , the test fails and the first derivative test has to be applied to study the nature of f(a).

Example:

Find the local maxima and local minima of the function f (x) = 2x³ - 21x² +36x - 20. Find also the local maximum and local minimum values.

Solution:

f '(x) = 6x² - 42x + 36

f '(x) = 0

x = 1 and x = 6 are the critical values

f ''(x) =12x - 42

If x =1, f ''(1) =12 - 42 = - 30 < 0

x =1 is a point of local maxima of f (x).

Maximum value = 2(1)³ - 21(1)² + 36(1) - 20 = -3

If x = 6, f ''(6) = 72 - 42 = 30 > 0

x = 6 is a point of local minima of f (x)

Minimum value = 2(6)³ - 21 (6)² + 36 (6)- 20

= -128

Absolute Maximum and Absolute Minimum Value of a Function

Let f (x) be a real valued function with its domain D.

(i) f(x) is said to have absolute maximum value at x = a if f(a) ³ f(x) for all x Î D.

(ii) f(x) is said to have absolute minimum value at x = a if f(a) £ f(x) for all x Î D.

The following points are to be noted carefully with the help of the diagram.

Let y = f (x) be the function defined on (a, b) in the graph.

(i) f (x) has local maximum values at

x = a₁, a₃, a₅, a₇

(ii) f (x) has local minimum values at

x = a₂, a₄, a₆, a₈

(iii) Note that, between two local maximum values, there is a local minimum value and vice versa.

(iv) The absolute maximum value of the function is f(a₇)and absolute minimum value is f(a).

(v) A local minimum value may be greater than a local maximum value.

Clearly local minimum at a₆ is greater than the local maximum at a₁.

Theorem 4

Let f be a continuous function on an interval I = [a, b]. Then, f has the absolute maximum value and f attains it at least once in I. Also, f has the absolute minimum value and attains it at least once in I.

Theorem 5

Let f be a differentiable function on I and let x₀ be any interior point of I. Then

(a) If f attains its absolute maximum value at x₀, then f ' (x₀)= 0

(b) If f attains its absolute minimum value at x₀, then f '(x₀) = 0.

In view of the above theorems, we state the following rule for finding the absolute maximum or absolute minimum values of a function in a given interval.

Step 1:

Find all the points where f ' takes the value zero.

Seep 2:

Take the end points of the interval.

Step 3:

At all the points calculate the values of f.

Step 4:

Take the maximum and minimum values of f out of the values calculated in step 3. These will be the absolute maximum or absolute minimum values.