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I have some sort of challenge questions/ homework from one of my math classes. It's been incredibly difficult and I would appreciate any help, thanks! Can someone please explain how to find the minimum/maximum values of these kind of questions?

Let x, y and z be positive real numbers. Find the minimum value of \((x + 2y + 4z) \left( \frac{4}{x} + \frac{2}{y} + \frac{1}{z} \right).\)
For positive real numbers x, y and z, find the minimum value of \(\frac{x^3 + 5y^3 + 25z^3}{xyz}\)

 May 20, 2022
 #1
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For the first one, you get the minimum when x = y = z.  So the minimum value is (x + 2x + 4x)/(4/x + 2/x + 1/x) = 49.

 

For the second one, you get the minimum when x = y = z.  So the minimum value is (x^3 + 5x^3 + 25x^3).(x^3) = 31.

 May 20, 2022
 #3
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Why?

Melody  May 20, 2022
 #2
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I thought the denominator in the agm formula needed to be the number of variables? 

 May 20, 2022

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