If-a-b-c-lt-0-and-abc-a-b-c-64-Then-find-min-of-P-2a-b-c-

Question Number 180873 by Shrinava last updated on 18/Nov/22

If a, b, c < 0 and abc (a + b + c) = 64

Then find \min of P = 2 a + b + c

Commented by mr W last updated on 18/Nov/22

p l e a s e c h e c k t h e q u e s t i o n . s i n c e

a, b, c < 0, P h a s m a x i m u m, n o t m i n .

o r y o u m e a n a, b, c > 0 ?

Commented by Shrinava last updated on 18/Nov/22

sorry dear professor > 0

Answered by mr W last updated on 18/Nov/22

d u e t o s y m m e t r y, P h a s e x t r e m u m

w h e n b = c = k, s a y

P = 2 a + b + c = 2 (a + k)

{a k}^{2} (a + 2 k) = 64

k^{2} a^{2} + 2 k^{3} a - 64 = 0

\Rightarrow a = \sqrt{k^{2} + \frac{64}{k^{2}}} - k

\Rightarrow P = 2 \sqrt{k^{2} + \frac{64}{k^{2}}} ⩾ 2 \sqrt{2 \times 8} = 8 = P_{m i n}

Commented by Shrinava last updated on 18/Nov/22

Thank you so much dear professor