Question

Let A, G, H be the arithmetic mean, geometric mean and harmonic mean respectively of two distinct positive real numbers. The roots of the quadratic equation Ax²−2Gx+H = 0 the interval
(A) (0,1)
(B) (1,2)
(C) (2,3)
(D) (3,4)

Answer 1

Answer:

        (A)  (0, 1)

Step-by-step explanation:

Let the two positive real numbers be a and b, so:

• A = (a + b) / 2
• G = √(ab)  ⇒  G² = ab
• H = 2/(1/a + 1/b) = 2ab/(a+b) = G²/A  ⇒  G² = AH

The discriminant of the quadratic equation Ax² - 2Gx + H is:

• Δ = (2G)² - 4AH = 4(G² - AH) = 0, since G² = AH.

So by the quadratic formula, the roots of the quadratic are actually a single repeated root:

• x = (2G ± √Δ)/(2A) = (2G)/(2A) = G/A.

By the AM-GM inequality, we have A > G (the inequality is strict since a and b are given to be distinct).  This gives...

• x = G/A < 1.

As both G and A are positive, x = G/A is positive, too.  Thus...

• 0 < x < 1,  or in other words,  x ∈ (0, 1).