If-a-4-b-4-c-4-d-4-16-prove-that-a-5-b-5-c-5-d-5-32-for-a-b-c-d-R-

Question Number 25025 by tawa tawa last updated on 01/Dec/17

If a^{4} + b^{4} + c^{4} + d^{4} = 16, prove that : a^{5} + b^{5} + c^{5} + d^{5} ⩽ 32

for a, b, c, d \in R

Answered by nnnavendu last updated on 02/Dec/17

ans

a^{4} + b^{4} + c^{4} + d^{4} = 16

puting b = c = d = 0

a^{4} + 0^{2} + 0^{2} + 0^{2} = 16

a^{4} = 16

a^{4} = 2^{4}

a = \mp 2

then

LHS -

a^{5} + b^{5} + c^{5} + d^{4}

{(\mp 2)}^{5} + 0^{2} + 0^{2} + 0^{2}

\mp 32 RHS

Commented by prakash jain last updated on 02/Dec/17

How do you prove that for any 4

a, b, c and d sum will remain less

that 32 . I mean the general case .

Commented by tawa tawa last updated on 10/Dec/17

Thanks for your help.

Commented by tawa tawa last updated on 10/Dec/17

God bless you sir

Answered by ajfour last updated on 02/Dec/17

Σ a^{4} = 16 \Rightarrow a^{4} ⩽ 16

\Rightarrow a - 1 ⩽ 1

Σ a^{5} = Σ (a - 1) a^{4} + Σ a^{4}

= (⩽ 16) + 16 ⩽ 32 .

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