let-x-y-gt-0-such-that-x-3-y-3-2-find-the-minimum-value-of-the-following-expression-P-2020x-2021y-

Question Number 159378 by HongKing last updated on 16/Nov/21

let x; y > 0 such that x^{3} + y^{3} = 2

find the minimum value of the

following expression :

P = 2020 x + 2021 y

Commented by mr W last updated on 16/Nov/21

P_{m a x} e x i s t s .

b u t f o r x, y > 0 P_{m i n} {d o e s n}^{'} t e x i s t .

o n l y f o r x, y ⩾ 0 P_{m i n} e x i s t s .

Commented by HongKing last updated on 16/Nov/21

yes my dear Ser, sorry find the maximum

Answered by mr W last updated on 16/Nov/21

f o r e x t r e m v a l u e o f P t h e l i n e

2020 x + 2021 y = P s h o u l d t a n g e n t t h e

c u r v e x^{3} + y^{3} = 2 .

2 x^{2} + 3 y^{2} \frac{d y}{d x} = 0

x^{2} - y^{2} \times \frac{2020}{2021} = 0

\Rightarrow y = \sqrt{\frac{2021}{2020}} x

x^{3} + \frac{2021}{2020} \sqrt{\frac{2021}{2020}} x^{3} = 2

x = \sqrt[3]{\frac{2}{1 + \frac{2021}{2020} \sqrt{\frac{2021}{2020}}}}

y = \sqrt{\frac{2021}{2020}} \sqrt[3]{\frac{2}{1 + \frac{2021}{2020} \sqrt{\frac{2021}{2020}}}}

P_{m a x} = (2020 + 2021 \sqrt{\frac{2021}{2020}}) \sqrt[3]{\frac{2}{1 + \frac{2021}{2020} \sqrt{\frac{2021}{2020}}}}

= 2020 \sqrt[3]{2 {(1 + \frac{2021}{2020} \sqrt{\frac{2021}{2020}})}^{2}}

\approx 4041.000062

Commented by HongKing last updated on 16/Nov/21

thank you so much my dear Ser perfect

Commented by mr W last updated on 16/Nov/21

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