Maxima & Minima

Definition:

A function f(x) is said to have a maximum value at x = a, if there exists a small number h, such that f(a) > both f(a – h) and f(a + h).

A function f(x) is said to have a minimum value at x = a, if there exists a small number

h, such that f(a) < both f(a – h) and f(a + h).

Note:

1. The maximum and minimum values of a function taken together are called its extreme values and the points at which the function attains the extreme values are called the turning points of the function.

2. A maximum or minimum value of a function is not necessarily the greatest or least value of the function in any finite interval. The maximum value is simply the greatest value in the immediate neighborhood of the maxima point or the minimum value is the least value in the immediate neighborhood of the minima point. In fact. there may be several maximum and minimum values of a function in an interval and a minimum value may be even greater than a maximum value.

3. It is seen from Fig. that maxima and minima values occur alternately.

(i) f(x) is maximum at x = a if f'(a) = 0 and f”(a) is negative.

 [i.e., f'(a) changes sign from positive to negative]

(ii) f(x) is minimum at x = a, if f’ (a) = 0 and f"(a) is positive.

[i.e., f'(a) changes sign from negative to positive)

4.

A maximum or a minimum value is a stationary value but a stationary value may neither be a maximum non a minimum value.

Procedure for finding maxima and minima

(i) Put the given function = f(x)

(ii) Find f’(x) and equate it to zero.

Solve this equation and let its roots be a, b, c, …

(iii) Find f"(x) and substitute in it by turns x = a, b, c, …

If f” (a) is negative, f(x) is maximum at x = a.

If f’’(a) is positive, f(x) is minima at x = a.

(iv) Sometimes f"(x) may be difficult to find out or f"(x) may be zero at x = a. In such cases, see if f’(x) changes sign from positive to negative as x passes through a, then f(x) is maximum at x = a.

If f’(x) changes sign from negative to positive as x passes through a, f(x) is minimum at x = a.

If f(x) does not change sign while passing through x = a, f(x) is neither maximum nor minimum at x = a.

MAXIMA - MINIMA OF FUNCTIONS OF TWO VARIABLES:

Let Z = f(x, y) be a given surface shown in the figure: 

Maxima:

Let Z = f(x, y) be any surface and let P(a, b) be any point on it then f(x, y) is called maximum at P(a, b) if f(a, b) > f(a + h, b + k) for all positive and negative values of h and k.

Minima:

Let Z = f(x, y) be any surface and let P(a, b) be any point on it then f(x, y) is called minimum at P(a, b) if f(a, b) < f(a + h, b + k)  for all positive and negative values of h and k.

Extremum:

The maximum or minimum value of the function f(x, y) at any point x = a and y = b is called the extremum value and the point is called the extremum point. 

Saddle Point:

It is a point where the function is neither maximum nor minimum. At this point, f is maximum in one direction while minimum in another direction. e.g. Consider Hyperbolic Paraboloid z = xy; since at origin (0, 0) function has neither maxima nor minima. So, origin is the saddle for Hyperbolic Paraboloid.

The Lagrange's conditions for maximum or minimum are:

Consider a function z = f(x,y) and let P(a, b) be any point on it, and let

(i) If rt – s² > 0 and r < 0, then f (x,y) has maximum value at (a,b).

(ii) If rt – s² > 0 and r > 0, then f (x, y) has minimum value at (a, b).

(iii) If rt – s² < 0, then f(x, y) has neither a minimum nor minimum i.e. (a, b) is a saddle point.

(iv) If rt – s² = 0, then the case fails and we need further investigations to calculate maxima or minima.