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Question Number 142200 by liberty last updated on 27/May/21

L e t p & q r e a l p o s i t i v e n u m b e r

w h a t t h e m i n i m u m o f {(\frac{p^{2}}{q^{2}} + \frac{q}{p})}^{3} .

Answered by MJS_new last updated on 27/May/21

let q = p t \land t > 0

{(\frac{p^{2}}{q^{2}} + \frac{q}{p})}^{3} = \frac{{(t^{3} + 1)}^{3}}{t^{6}}

\frac{d}{d t} [\frac{{(t^{3} + 1)}^{3}}{t^{6}}] = 0 \land t > 0 \Rightarrow t = \sqrt[3]{2} \Rightarrow \min is \frac{27}{4}

Answered by mitica last updated on 27/May/21

{(\frac{p^{2}}{q^{2}} + \frac{q}{2 p} + \frac{q}{2 p})}^{3} \overset{a m - g m}{⩾} {(3 \sqrt[3]{\frac{p^{2}}{q^{2}} \cdot \frac{q}{2 p} \cdot \frac{q}{2 p}})}^{3} = \frac{27}{4}

,, =^{″} f o r \frac{p^{2}}{q^{2}} = \frac{q}{2 p} \Leftrightarrow p = q \sqrt[3]{2} \Rightarrow m i n = \frac{27}{4}

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