Application of DerivativeHard
Question
If a1, a2 ........ an ∈ R+ and a1, a2 .......an = 1 then the least value of (1 + a1 + a21 ) (1 + a2 + a22 )....(1 + an + an2 ) is -
Options
A.3n
B.n3n
C.33n
D.data inadequate
Solution
Using AM ≥ GM,
≥ (1. a1a21)1/3 ⇒ 1 + a1 + a12 ≥ 3a1
⇒ 1 + a2 + a22 ≥ 3a2 ⇒ ....... 1 + an + a2n ≥ 3an
Multiplying these,
(1 + a + a21) .... (1 + an + a2n) ≥ 3n (a1a2a1 ... an) = 3n .1
≥ (1. a1a21)1/3 ⇒ 1 + a1 + a12 ≥ 3a1⇒ 1 + a2 + a22 ≥ 3a2 ⇒ ....... 1 + an + a2n ≥ 3an
Multiplying these,
(1 + a + a21) .... (1 + an + a2n) ≥ 3n (a1a2a1 ... an) = 3n .1
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