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Home / 2y + \frac{1}{2y} \geq 2\sqrt{2y \cdot \frac{1}{2y}} = 2

2y + \frac{1}{2y} \geq 2\sqrt{2y \cdot \frac{1}{2y}} = 2

February 22, 2026

$2y + \frac{1}{2y} \geq 2\sqrt{2y \cdot \frac{1}{2y}} = 2$

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We minimize $ 2y + \frac{1}{2y} + 2 $. Let:
h(y) = 2y + \frac{1}{2y}
Using the AM-GM inequality:
Equality holds when $ 2y = \frac{1}{2y} \Rightarrow 4y^2 = 1 \Rightarrow y = \frac{1}{2} $.
g(x) = h(y) + 2 = 2 + 2 = 4
Therefore, the minimum value of $ g(x) $ is:

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