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Two positive numbers $x$ and $y$ vary in such a way that

$\ 128x^2-16x^2y+1=0$

Find the minimum value of $(x+y)$.

The answer is 35/4, how do I get the answer?

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    $\begingroup$ Lagrange Multipliers is one way. Still, are you sure the $1$ is not a $-1$? $\endgroup$ – Gary. Aug 22 '15 at 16:00
  • $\begingroup$ Yea, I checked the question paper again. Also, I'm only in a Secondary School (in Singapore) so I haven't learnt all these methods yet. I've only learnt a fair bit of calculus so that's all I can use to solve the question. $\endgroup$ – Sheow Boon Aug 22 '15 at 16:12
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$$y=8+\dfrac1{16x^2}$$

$$\implies x+y=x+8+\dfrac1{16x^2}$$

As $x>0,$ using AM, GM inequality $$x+\dfrac1{16x^2}=\dfrac x2+\dfrac x2+\dfrac1{16x^2}=\dfrac{\dfrac x2+\dfrac x2+\dfrac1{16x^2}}3\ge\sqrt[3]{\dfrac x2\cdot\dfrac x2\cdot\dfrac1{16x^2}}=\dfrac14$$

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  • $\begingroup$ Is there another method that only uses calculus? I'm only in a secondary school (Equivalent of a high school) so I haven't learnt this method yet. Looking at the question, I'm fairly sure I need to use calculus to solve this question. Is there such a method? $\endgroup$ – Sheow Boon Aug 22 '15 at 16:21
  • $\begingroup$ @SheowBoon, Use en.wikipedia.org/wiki/Second_derivative_test on $$f(x)=x+8+\dfrac1{16x^2}$$ $\endgroup$ – lab bhattacharjee Aug 22 '15 at 16:22

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