Paper 4: Fundamentals of Business Mathematics & Statistic

3.46 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Since dydx is continuous function of x it can change sign only after passing through zero value.

Thus dydx = 0.

Hence for a function y = f(x) to attain maximum value at x = 1.

(i) dydx = 0, (ii) dydxchanges sign from + ve to – ve at x = a, i.e., is a decreasing function of x and so

2

2

dy 0.

dx <

(b) If again a continuous function y = f(x) is minimum at x = a, then by definition it is decreasing just before

x = a and then increasing just after x = a, i.e., its derivative dydx is – ve just before x = a and

+ ve just after x = a. This means dydx changes sign from – ve to + ve values. A continuous function dydx can

change sign only after passing through zero value, so dydx = 0.

Hence for a continuous function y = f(x) to attain a minimum value at x = a,

(i) dydx = 0, (ii) dydxchanges sign from – ve at + ve at x = a, i.e., dydx is an increasing function of x hence

2

2

d y 0.

dx >

Summary :

For a function y = f(x) to attain a maximum point at x = a,

(i) ddxy=0, (ii)

2

2

dy 0,

dx < and

for a minimum point

(i) ddxy= 0 (ii)

2

2

dy 0.

dx >

Conditions for Maximum and Minimum : Necessary Condition. If a function f(x) is maximum or minimum at

a point x = b and if f′(b) exists then f′ (b) = 0.

Sufficient Condition : If b is a point in an interval where f(x) is defined and if f ′ (b) = 0 and f′′ (b) ≠ 0, then f(b)

is maximum if f ′′ (b) <0 and is minimum if f ′′ (b) > 0. (The proof is not shown at present).

Definition :

If in a function y = f(x), for continuous increasing value of x, y increases upto a certain value and then

decreases, then this value of y is said to be the maximum value. If again y decreases upto a certain value

and then increases for continuous increasing value of x, then this value y is said to be minimum value. The

points on the curve y = f(x). which separate the function from its increasing state to decreasing state or vice

versa are known as turning points on the curve. From these turning points the curve may attain the extreme

values. (i.e., maximum or minimum).