Positive x , y , z are in A.P. The minimum of 2 y − x x + y + 2 y − z y + z is less than which listed values?

Since x + z = 2 y , the expression becomes ( x + y ) / z + ( y + z ) / x = 1 + 2 3 ( x / z + z / x ) ≥ 1 + 3 = 4 . Thus its minimum 4 is less than 6 and 8 only.

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MathematicsSequence & SeriesMeans / inequalityMedium2 minQB

MathematicsMediummultiple choice

Positive $x, y, z$ are in A.P. The minimum of $\frac{x + y}{2 y - x} + \frac{y + z}{2 y - z}$ is less than which listed values?

Options:(select one or more)

Answer:

B, C

Solution:

Since $x + z = 2 y$ , the expression becomes $(x + y) / z + (y + z) / x = 1 + \frac{3}{2} (x / z + z / x) \geq 1 + 3 = 4$ . Thus its minimum 4 is less than 6 and 8 only.

Stream:JEESubject:MathematicsTopic:Sequence & SeriesSubtopic:Means / inequality

⏱ 2mℹ️ Source: QB

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