Given that xy = 1/2 and both x and y are nonnegative real numbers, find the minimum value of 4x + 9y.
+14995 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.
!
+14995 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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