Find-the-maximum-value-of-the-function-f-x-y-x-2-y-2-z-2-subject-to-the-condition-that-x-2-y-2-z-2-c-2-where-c-is-the-constant-Thank-you-in-advance-

Question Number 203527 by Mastermind last updated on 21/Jan/24

Find the maximum value of the function

f (x, y) = x^{2} y^{2} z^{2} subject to the condition that

x^{2} + y^{2} + z^{2} = c^{2}, where c is the constant .

Thank you in advance

Answered by aleks041103 last updated on 21/Jan/24

x^{2} = X

y^{2} = Y

z^{2} = Z

c^{2} = C

m a x o f X Y Z i f X + Y + Z = C, X, Y, Z, C > 0

g e o m . m e a n ⩽ a r i t h m . m e a n

\Rightarrow \sqrt[3]{X Y Z} ⩽ \frac{X + Y + Z}{3} = \frac{C}{3} (e q u a l i t y i f X = Y = Z)

\Rightarrow X Y Z ⩽ \frac{C^{3}}{27}

\Rightarrow m a x f (x, y, z) = x^{2} y^{2} z^{2} i f x^{2} + y^{2} + z^{2} = c^{2}

i s m a x (f) = \frac{c^{6}}{27} f o r ∣ x ∣=∣ y ∣=∣ z ∣= \frac{∣ c ∣}{\sqrt{3}}

Commented by Mastermind last updated on 21/Jan/24

Thank you so much but could you please

help me with full details

Thank you once again

Commented by aleks041103 last updated on 21/Jan/24

b a s i c a l l y w e u s e t h e i n e q u a l i t y

G M ⩽ A M

w h e r e G M i s t h e g e o m e t r i c m e a n

a n d A M i s t h e a r i t h m e t i c m e a n

Answered by AST last updated on 21/Jan/24

c^{2} = x^{2} + y^{2} + z^{2} ⩾ 3 \sqrt[3]{x^{2} y^{2} z^{2}} \Rightarrow x^{2} y^{2} z^{2} ⩽ \frac{c^{6}}{27}

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