If-a-b-c-d-1-2-then-prove-that-a-b-c-d-e-1-a-1-b-1-c-1-d-1-e-28-When-equality-holds-

Question Number 178818 by Shrinava last updated on 21/Oct/22

If a, b, c, d \in [1, 2] then prove that :

(a + b + c + d + e) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e}) ⩽ 28

When equality holds ?

Answered by mr W last updated on 22/Oct/22

A = a + b + c + d + e

B = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e}

Φ = A \times B

w e s e e, w h e n w e m a k e A l a r g e r, w e

m a k e B s m a l l e r i n t h e s a m e t i m e .

t o m a k e A l a r g e r, w e s h o u l d m a k e

a s m a n y n u m b e r s a s p o s s i b l e e q u a l

t o 2 . b u t t o m a k e B l a r g e r, w e s h o u l d

m a k e a s m a n y n u m b e r s a s p o s s i b l e

e q u a l t o 1 .

w e h a v e 5 n u m b e r s . {l e t}^{'} s s a y n n u m b e r s

e q u a l 2 a n d 5 - n n u m b e r s e q u a l 1 .

A = n \times 2 + (5 - n) \times 1 = n + 5

B = n \times \frac{1}{2} + (5 - n) \times \frac{1}{1} = 5 - \frac{n}{2}

Φ = (n + 5) (5 - \frac{n}{2}) = \frac{50 + 5 n - n^{2}}{2}

= \frac{225}{8} - \frac{1}{2} {(n - \frac{5}{2})}^{2}

s i n c e n i s i n t e g e r, n \neq \frac{5}{2} .

Φ_{m a x} i s a t n = 2 o r 3 .

Φ_{m a x} = \frac{225}{8} - \frac{1}{2} \times {(\frac{1}{2})}^{2} = \frac{224}{8} = 28

i . e . Φ ⩽ 28 . t h e e q u a l i t y h o l d s w h e n

2 n u m b e r s e q u a l t o 2 a n d 3 n u m b e r s

e q u a l t o 1, o r w h e n 3 n u m b e r s e q u a l

t o 2 a n d 2 u m b e r s e q u a l t o 1 .

Commented by Tawa11 last updated on 22/Oct/22

Great sir

Commented by Shrinava last updated on 23/Oct/22

cool dear professor thank you so much