Question

If a, b, c, d are positive real numbers such that a + b + c + d =2, then M=(a +b) (c + d) satisfies the relation

(a) 0 < M < 1 (b) 1 < M < 2 (c) 2 < M < 3 (d) 3 < M < 4

Answer 1

The given numbers are positive and real and

a + b + c + d = 2

Now, M = (a + b) (c + d) ......... (1)

If we take a + b large i.e. say 1.9, then obviously c + d will be 0.1 as the total sum of a, b, c, d is 2 and when we multiply these two, the product will be 0.19 which is less than 1.

The maximum value of the product (a + b) (c + d) is possible when we take (a + b) = 1 and (c + d) = 1.

Since, the numbers are given positive, so M ≥ 0.

Hence, 0 ≤ M ≤ 1 is true.