1.

(i) Let f(x) be a function with domain D \subset R . Then f(x) is said to attain the maximum value at a point a \in D if f(x) \le f(a) for all x \in D .

In such a case, a is called the point of maximum and f(a) is known as the maximum value or the greatest value or the absolute maximum value of f(x) .

(ii) Let f(x) be a function with domain D \subset R . Then f(x) is said to attain the minimum value at a point a \in D if f(x) \ge f(a) for all x \in D .

In such a case, a is called the point of minimum and f(a) is known as the minimum value or the least value or the absolute minimum value of f(x) .

(iii) A function f(x) is said to attain a local maximum at x = a if there exists a neighbourhood (a - \delta, a + \delta) of a such that

f(x) \le f(a) for all x \in (a - \delta, a + \delta) , x \ne a ,

or

f(x) - f(a) < 0 for all x \in (a - \delta, a + \delta) , x \ne a .

In such a case, f(a) is called the local maximum value of f(x) at x = a .

(iv) A function f(x) is said to attain a local minimum at x = a if there exists a neighbourhood (a - \delta, a + \delta) of a such that

f(x) \ge f(a) for all x \in (a - \delta, a + \delta) , x \ne a ,

or

f(x) - f(a) > 0 for all x \in (a - \delta, a + \delta) , x \ne a .

The value of the function at x = a , i.e., f(a) , is called the local minimum value of f(x) at x = a .

The points at which a function attains either local maximum values or local minimum values are known as extreme points or turning points.
Both local maximum and local minimum values are called the extreme values of f(x) .

Thus, a function attains an extreme value at x = a if f(a) is either a local maximum value or a local minimum value.
Consequently, at an extreme point a , the expression f(x) - f(a) keeps the same sign for all values of x in a deleted neighbourhood of a .

2.

A necessary condition for f(a) to be an extreme value of a function f(x) is that

f'(a) = 0 , in case it exists.

However, a function may attain an extreme value at a point without being differentiable there. For example, the function f(x) = |x| attains its minimum value at the origin even though it is not differentiable at x = 0 .

This condition is only a necessary condition, not sufficient. That is, f'(a) = 0 does not necessarily imply that x = a is an extreme point.

There are functions for which the derivative vanishes at a point but the function does not attain an extreme value there. For example, for the function f(x) = x^3 , we have f'(0) = 0 , but at x = 0 the function does not attain an extreme value.

Geometrically, this condition means that the tangent to the curve y = f(x) at a point where the ordinate is maximum or minimum is parallel to the x -axis.

As discussed earlier, all values of x for which f'(x) = 0 do not necessarily give extreme values.

The values of x for which f'(x) = 0 are called stationary values or critical values of x , and the corresponding values of f(x) are called stationary values or turning values of f(x) .

3. First Derivative Test for Local Maxima and Minima

Let f(x) be a function differentiable at x = a .

(a) x = a is a point of local maximum of f(x) if:

f'(a) = 0 , and
f'(x) changes sign from positive to negative as x passes through a ,

i.e.,

f'(x) > 0 for every point in the left neighbourhood (a - \delta, a) of a , and

f'(x) < 0 for every point in the right neighbourhood (a, a + \delta) of a .

(b) x = a is a point of local minimum of f(x) if:

f'(a) = 0 , and
f'(x) changes sign from negative to positive as x passes through a ,

i.e.,

f'(x) < 0 for every point in the left neighbourhood (a - \delta, a) of a , and

f'(x) > 0 for every point in the right neighbourhood (a, a + \delta) of a .

(c) If f'(a) = 0 , but f'(x) does not change sign (that is, f'(a) has the same sign in the complete neighbourhood of a ), then a is neither a point of local maximum nor a point of local minimum.

4. Higher Order Derivative Test

Let f be a differentiable function on an interval I and let c be an interior point of I such that:

f'(c) = f''(c) = f'''(c) = \dots = f^{(n-1)}(c) = 0 , and
f^{(n)}(c) exists and is non-zero.

Then:

If n is even and f^{(n)}(c) < 0 , then x = c is a point of local maximum.
If n is even and f^{(n)}(c) > 0 , then x = c is a point of local minimum.
If n is odd, then x = c is neither a point of local maximum nor a point of local minimum.

Steps to Find Local Maxima / Minima

Step I: Find f'(x) .

Step II: Put f'(x) = 0 and solve the equation for x . Let c_1, c_2, \dots, c_n be the roots of this equation. These are called stationary values (or critical values) of x . These are the possible points where the function may attain local maximum or minimum.

Step III: Find f''(x) . Consider x = c_1 .

If f''(c_1) < 0 , then x = c_1 is a point of local maximum.
If f''(c_1) > 0 , then x = c_1 is a point of local minimum.
If f''(c_1) = 0 , then find f'''(x) and substitute x = c_1 .

If f'''(c_1) \neq 0 , then x = c_1 is neither a point of local maximum nor local minimum and is called a point of inflection.

If f'''(c_1) = 0 , then find f^{IV}(x) and substitute x = c_1 .

If f^{IV}(c_1) < 0 , then x = c_1 is a point of local maximum.
If f^{IV}(c_1) > 0 , then x = c_1 is a point of local minimum.

If f^{IV}(c_1) = 0 , then find f^{V}(x) and continue similarly.

Similarly, the values c_2, c_3, \dots may be tested.

5. Properties of Maxima and Minima

(i) If f(x) is continuous in its domain, then at least one maximum and one minimum must lie between two equal values of x .

(ii) Maxima and minima occur alternately; that is, between two maxima there is one minimum and vice-versa.

(iii) If f(x) \to \infty as x \to a or b and f'(x) = 0 only for one value of x (say at x = c ) between a and b , then f(c) is necessarily the minimum and the least value.

If f(x) \to -\infty as x \to a or b , then f(c) is necessarily the maximum and the greatest value.

6. Absolute Maximum and Minimum on a Closed Interval

The maximum and minimum values of a function defined on a closed interval may be obtained using the following steps.

Let y = f(x) be a function defined on [a, b] .

Step I: Find \frac{dy}{dx} = f'(x) .

Step II: Put f'(x) = 0 and find the values of x . Let c_1, c_2, \dots, c_n be the values of x .

Step III: Take the maximum and minimum values out of f(a), f(c_1), f(c_2), \dots, f(c_n), f(b) .

The maximum and minimum values obtained in Step III are respectively the largest (absolute maximum) and the smallest (absolute minimum) values of the function.

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