Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.53

Example 112 :The sum of two numbers is 12. Find the maximum value of their product.
Solution:
Let the two numbers be x and y, so that x + y = 12.
Product (P) = xy = x (12 – x) = 12x – x^2

dp 12 2x.
dx= −
For product max.

dp 0

dx= i.e., 12 – 2x = 0 or, x = 6. Agian

2

2

d p 2 0,max.

dx

= − <

∴ reqd. product = x (12 – x) = 6 (12 – 6) = 6.6 = 36
Example 113 :A wire of length 16cm is to form a rectanglle. Find the dimensions of a rectangle so that it has
maximum area.
Solution:
Let length be x, breadth be y so that
2x + 2y = 16 or, x + y = 8 or, y = 8 – x
Area (A) = length × breadth = xy = x (8 – x)

dA 8 2x

dx= −. For max. area we have

dA 0

dx= , or 8 – 2x = 0 or, x = 4;

2

2

dy 2 0,max.

dx = − <

For x = 4, y = 8 – 4 = 4.
So the area is maximum for length = breadth is 4 cm i.e., rectangle is a square.

SELF EXAMINATION QUESTION

1. Find the which values of x the following functions are maximum and minimum:
   (i) x (12 – 2x)^2 (ii) x^3 – 3x^2 – 9x + 5

(iii) x3 – 6x2 + 9x – 8 (iv)

2

2

x x 1

x x 1

+ +

− +

(v) x^3 – 9x^2 + 24x – 12. [Ans. (i) 2, 6; (ii) –1, 3; (iii) 1, 3; (iv) 1, –1 (v) 2, 4]

2. Find the maximum and minimum values of the above example.

[Ans. (i) 128, 0 (ii) 10, –22, (iii) –4, –8, (iv)^313 , (v) 8, 4]

3. Show that the maximum value of x+^1 x is less than its minimum value