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Range of values of D= A-G ; for A is the A.M and G is GM of 'a' and 'b'

Range of values of D= A-G ; for A is the A.M and G is GM of 'a' and 'b'

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Grade:12

1 Answers

Aditya Gupta
2081 Points
5 years ago
we see that D= (a+b – 2√(ab))/2 = (√a – √b)^2/2. now when a=b, all the options are satisfies. so we assume that a is not equal to b.
so, 2D/(a – b)^2= [(√a – √b)/(a –  b)]^2= 1/(√a + √b)^2
now it is given that a is greater than or equal to b.
so that (√a + √b)^2 is greater than 4a.
to see why, consider the term (√a + √b)^2 – 4a= a+b+2√(ab) – 4a= (b – a) + 2√a(√b – √a), which is obviously greater than zero as both (b – a) and 2√a(√b – √a) are greater than zero. hence the conclusion (√a + √b)^2 is greater than 4a.
or 1/(√a + √b)^2 is less than 1/4a. but 1/(√a + √b)^2=2D/(a – b)^2 
so 2D/(a – b)^2 is less than 1/4a.
or D is less than (a – b)^2/8a.
similarly we can prove that (√a + √b)^2 is less than 4b. using which it is easy to show that D is greater than (a – b)^2/8b.
kindly approve :)

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