if-2x-3-3y-3-200-x-2-y-2-max-

Question Number 148027 by mathdanisur last updated on 25/Jul/21

i f 2 x^{3} + 3 y^{3} = 200 \Rightarrow {(x^{2} y^{2})}_{m a x} = ?

Commented by mr W last updated on 25/Jul/21

200 = 2 x^{3} + 3 y^{3} ⩾ 2 \sqrt{(2 x^{3}) (3 y^{3}}) = 2 \sqrt{6} {(x^{2} y^{2})}^{\frac{3}{4}}

\Rightarrow x^{2} y^{2} ⩽ {(\frac{200}{2 \sqrt{6}})}^{\frac{4}{3}} = \frac{500}{\sqrt[3]{45}} = m a x

Commented by mathdanisur last updated on 25/Jul/21

T h a n k y o u S i r

Commented by mathdanisur last updated on 25/Jul/21

S e r, y o u u s e d t h e c o n d i t i o n t h a t t h e

v a r i a b l e s a r e n o t n e g a t i v e, b u t t h e

c o n d i t i o n i s n o t g i v e n

Commented by mr W last updated on 25/Jul/21

f o r l o c a l m a x i m u m, b o t h x a n d y a r e

p o s i t i v e .

i n f a c t t h i s q u e s t i o n i s n o t a g o o d o n e,

s i n c e (x^{2} y^{2}) h a s n e i t h e r a m a x i m u m

n o r a m i n i m u m .

Answered by Olaf_Thorendsen last updated on 25/Jul/21

2 x^{3} + 3 y^{3} = 200

x = 200^{1 / 3} \frac{\cos^{2 / 3} t}{2^{1 / 3}}, y = 200^{1 / 3} \frac{\sin^{2 / 3} t}{3^{1 / 3}}

x^{2} y^{2} = \frac{200^{4 / 3}}{2^{2 / 3} 3^{2 / 3}} \cos^{4 / 3} t . \sin^{4 / 3} t

x^{2} y^{2} = \frac{200^{4 / 3}}{2^{2 / 3} 3^{2 / 3} 2^{4 / 3}} \sin^{4 / 3} (2 t)

x^{2} y^{2} = \frac{2^{2} . 5^{8 / 3}}{3^{2 / 3}} \sin^{4 / 3} (2 t)

\frac{d}{d t} (x^{2} y^{2}) = \frac{2^{2} . 5^{8 / 3}}{3^{2 / 3}} (2 . \frac{4}{3} . \cos (2 t) \sin^{1 / 3} (2 t))

\frac{d}{d t} (x^{2} y^{2}) = \frac{2^{5} . 5^{8 / 3}}{3^{5 / 3}} (\cos (2 t) \sin^{1 / 3} (2 t))

\frac{d}{d t} (x^{2} y^{2}) = 0

\Leftrightarrow 2 t = \pm \frac{k π}{2} + 2 k π or 2 t = \pm k π

∙ For t = \frac{π}{4} :

x = 200^{1 / 3} \frac{\cos^{2 / 3} \frac{π}{4}}{2^{1 / 3}} = 2^{1 / 3} 5^{2 / 3}

y = 200^{1 / 3} \frac{\sin^{2 / 3} \frac{π}{4}}{3^{1 / 3}} = \frac{2^{2 / 3} 5^{2 / 3}}{3^{1 / 3}}

x^{2} y^{2} = {(2^{1 / 3} 5^{2 / 3})}^{2} {(\frac{2^{2 / 3} 5^{2 / 3}}{3^{1 / 3}})}^{2} = \frac{2^{2} 5^{8 / 3}}{3^{2 / 3}}

Commented by mr W last updated on 25/Jul/21

p l e a s e c h e c k b e f o r e s a y i n g!

b u t 50 (25^{\frac{1}{3}}) i s w r o n g!

a n d \frac{2^{2} 5^{8 / 3}}{3^{2 / 3}} i s r i g h t!

Commented by mathdanisur last updated on 25/Jul/21

T h a n k y o u S e r

B u t a n s w e r : 50 (25^{\frac{1}{3}})

Commented by mathdanisur last updated on 25/Jul/21

T h a n k y o u S e r