x-5-y-3-5-x-3-y-139-x-y-R-gt-0-find-maximum-and-minimum-of-x-y-xy-

Question Number 153513 by yeti123 last updated on 08/Sep/21

(\frac{x}{5} + \frac{y}{3}) (\frac{5}{x} + \frac{3}{y}) = 139, \forall x, y \in R_{> 0}

find maximum and minimum of \frac{x + y}{\sqrt{x y}}

Answered by liberty last updated on 08/Sep/21

\Rightarrow \frac{x + y}{\sqrt{x y}} = \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = \sqrt{\frac{5 λ}{3}} + \sqrt{\frac{3}{5 λ}}

f r o m c o n d i t i o n

2 + \frac{3 x}{5 y} + \frac{5 y}{3 x} = 139

\frac{3 x}{5 y} + \frac{5 y}{3 x} = 137 s e t \frac{3 x}{5 y} = λ \Rightarrow \frac{x}{y} = \frac{5 λ}{3}

λ + \frac{1}{λ} = 137 \Rightarrow λ^{2} - 137 λ + 1 = 0

λ = \frac{137 \pm \sqrt{(137 + 2) (137 - 2)}}{2}

λ = \frac{137 \pm 136.98}{2} \to {\begin{cases} λ_{1} = 136.99 \\ λ_{2} = 0.007 \end{cases}

f (λ_{1}) = \sqrt{\frac{5 \times 136.99}{3}} + \sqrt{\frac{3}{5 \times 136.99}}

\approx 15.176 (m a x)

f (λ_{2}) = \sqrt{\frac{5 \times 0.007}{3}} + \sqrt{\frac{3}{5 \times 0.07}}

\approx 9.366 (m i n)

Commented by yeti123 last updated on 08/Sep/21

ok, thanks

Commented by MJS_new last updated on 08/Sep/21

you should round significant figures

λ_{1} \approx 136.99

λ_{2} \approx . 0072997

\Rightarrow

f (λ_{1}) \approx 15.176

f (λ_{2}) \approx 9.1765

as you can see your path is ok but your 2^{nd}

result is not \dots

Commented by liberty last updated on 08/Sep/21

Commented by liberty last updated on 08/Sep/21

o y e s \dots

Answered by MJS_new last updated on 08/Sep/21

strange question \dots

(\frac{x}{5} + \frac{y}{3}) (\frac{5}{x} + \frac{3}{y}) = 139 \Rightarrow y = \frac{411 \pm 9 \sqrt{2085}}{10} x

\Rightarrow \frac{x + y}{\sqrt{x y}} = \pm 3 + \frac{4 \sqrt{2085}}{15}

\Rightarrow there are only 2 values for x > 0

\min = - 3 + \frac{4 \sqrt{2085}}{15}

\max = 3 + \frac{4 \sqrt{2085}}{15}

Commented by liberty last updated on 08/Sep/21

3 + \frac{4 \sqrt{2085}}{15} = 15.176

Commented by MJS_new last updated on 08/Sep/21

no .

\approx 15.176

Commented by liberty last updated on 08/Sep/21

��

Commented by yeti123 last updated on 08/Sep/21

ok, I get it . thanks

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