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If a x^2+b/xgeqc for all positive x wher...

If `a x^2+b/xgeqc` for all positive `x` where `a >0` and `b >0,` show that `27 a b^2geq4c^3dot`

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To solve the problem, we need to show that \( 27ab^2 \geq 4c^3 \) given the inequality \( ax^2 + \frac{b}{x} \geq c \) for all positive \( x \), where \( a > 0 \) and \( b > 0 \). ### Step-by-Step Solution: 1. **Define the Function:** Let \( f(x) = ax^2 + \frac{b}{x} - c \). We need to show that \( f(x) \geq 0 \) for all positive \( x \). 2. **Find the Derivative:** ...
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