If x, y and z are positive numbers and x + y + z = 1, then find the least value of
1/x + 1/y + 1/z.

Question

If x, y and z are positive numbers and x + y + z = 1, then find the least value of
 1/x + 1/y + 1/z.

Accepted Answer

Given x, y, z > 0 and x + y + z = 1, we want to minimize 1/x + 1/y + 1/z.

Using the AM-HM inequality for three positive numbers x, y, z:

Arithmetic Mean (AM) = (x + y + z)/3 = 1/3
Harmonic Mean (HM) = 3 / (1/x + 1/y + 1/z)

By AM-HM inequality:
AM ≥ HM
=> (1/3) ≥ 3 / (1/x + 1/y + 1/z)
=> 1/x + 1/y + 1/z ≥ 9

Equality holds when x = y = z = 1/3.

Therefore, the least value of 1/x + 1/y + 1/z is 9.

Answer

Answer

Answer