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# minimum value

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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.

Oct 13, 2020

#1
+10820
+2

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

Hello Guest!

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

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

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

$$x=1.06066$$

$$y=0.47140$$

$$4x+9y=8.48528$$

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

!

Oct 13, 2020
#2
+10820
0

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.

!

Oct 13, 2020