If x,y and z are positive numbers and x + y + z = 1, then the least value of 1/x + 1/y + 1/z is:
Question:
If x,y and z are positive numbers and x + y + z = 1, then the least value of 1/x + 1/y + 1/z is:
If x,y and z are positive numbers and x + y + z = 1, then the least value of 1/x + 1/y + 1/z is:
Options
Answer: (3) 9
Explanation:
Given x, y, z > 0 and x + y + z = 1 We want the least value of: 1/x + 1/y + 1/z Using AM–HM inequality: (x + y + z) / 3 ≥ 3 / (1/x + 1/y + 1/z) Substitute x + y + z = 1: 1/3 ≥ 3 / (1/x + 1/y + 1/z) Invert both sides: 1/x + 1/y + 1/z ≥ 9 Equality occurs when x = y = z = 1/3. Least value = 9
Explanation:
Given x, y, z > 0 and x + y + z = 1 We want the least value of: 1/x + 1/y + 1/z Using AM–HM inequality: (x + y + z) / 3 ≥ 3 / (1/x + 1/y + 1/z) Substitute x + y + z = 1: 1/3 ≥ 3 / (1/x + 1/y + 1/z) Invert both sides: 1/x + 1/y + 1/z ≥ 9 Equality occurs when x = y = z = 1/3. Least value = 9
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