minimum value

Given that xy = 1/2  and both x and y are nonnegative real numbers, find the minimum value of 4x + 9y.

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$xy=\frac{1}{2}$       $y=\frac{1}{2x}$

$f(x)=4x+9y\\ f(x)=4x+\frac{9}{2x}\\ f(x)=4x+4.5x^{-1}$

$\color{blue}\frac{df(x)}{dx}=4-4.5x^{-2}=0\\ \frac{4.5}{x^2}=4\\ x^2=\frac{4.5}{4}$

$x=1.06066$

$y=0.47140$

$4x+9y=8.48528$

The minimum value of 4x + 9y is 8.48528.

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Given that xy = 1/2  and both x and y are nonnegative real numbers, find the minimum value of 4x + 9y.

 $xy=\frac{1}{2}$       $x=\frac{1}{2y}$

$f(y)=4x+9y\\ f(y)=\frac{2}{y}+9y=2y^{-1}+9y$

$\color{blue}\frac{df(y)}{dy}=-2y^{-2}+9=0\\ \frac{2}{y^2}=9\\ y^2=\frac{2}{9}$

$y=0.42140\\ x=1.06066$

       ⇓

$4x+9y=8.48528$

q.e.d.

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