FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.45Again x + y – 10 = 0 .... (iii) solving (i), (ii), (iii) we get x = 5, y = 5, l = – 10. At x = 5, y = 5, the given
function f(x, y) has minimum value and the value is 5^2 + 5^2 = 50.
3.4.6 MAXIMUM AND MINIMUM
A function f(x) is said to be maximum at x = a if f(a) is greater than every other value of f(x) in the immediate
neighbourhood of x = a (i.e., f(x) ceases to increase but begins to increase at x = a. Similarly the minimum
value of f(x) will be that value at x = b which is less than other values in the immediate neighbourhood of x
= b. [i.e., f(x) ceases to decrease but begins to increase at x = b]
The above figure represents graphically a continuous function f(x). The function has a maximum values at
P 1 , P 2 , P 3 and also minimum values at Q 1 , Q 2. For P 2 , abscissa is OL 2 , ordinate is P 2 L 2. Similarly OR 1 and R 1 Q 1 are
the respective abscissa and ordinate to Q 1. In the immediate neighbourhood of L 2 , we may get a range of
M 1 L 2 M 2 (on either side of L 2 ) such that for every value of x within that range (expect at L 2 ), the value f(x) is
less than P 2 L 2 (i.e., the value at L 2 ),. Hence we can now show that f(x) is maximum at x = OL 2. In the same
way we may find a neighbourhood N 1 R 1 N 2 or R 1 so that for every value of x within the range (expect at R 1 )
the value of f(x) is greater than at R 1. So the function is minimum at R 1.
The ordinate P 2 L 2 should not necessarily be bigger than the ordinate R 1 R 1.
Features regarding Maximum and Minimum : (i) Function may have several maximum and minimum
values in an interval (as shown in Fig. Above).
(ii) Maximum and minimum values of a function occur alternatively (for clear idea see Fig above).
(iii) At some point the maximum value may be less than the minimum value (i.e., Fig., P 2 L 2 < Q 2 R 2 ).
(iv) In the graph of the function maxima are like mountain tops while minima are like valley bottoms.
(v) The points at which a function has maximum or minimum value are called turning points and the
maximum and minimum values are known as extreme values, or extremum or turning values.
(vi) The values of x for which f(x) = 0 are often called critical values.
Criteria for Maximum and Minimum : (a) If a continuous function y = f(x) is maximum at a point x = a (say),
then by definition, it is an increasing function for values of x just before x = a and a decreasing function for
values of x just after x = a, i.e., its derivative dydx is positive before x = a and negative after x = a. This means
at x = a, dydx changes sign from + ve to – ve.