Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

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 ∈ R

$$\mathrm{If}\:\:\:\:\mathrm{a}^{\mathrm{4}} \:+\:\mathrm{b}^{\mathrm{4}} \:+\:\mathrm{c}^{\mathrm{4}} \:+\:\mathrm{d}^{\mathrm{4}} \:=\:\mathrm{16},\:\:\mathrm{prove}\:\mathrm{that}:\:\:\mathrm{a}^{\mathrm{5}} \:+\:\mathrm{b}^{\mathrm{5}} \:+\:\mathrm{c}^{\mathrm{5}} \:+\:\mathrm{d}^{\mathrm{5}} \:\leqslant\:\mathrm{32} \\ $$$$\mathrm{for}\:\:\mathrm{a},\:\mathrm{b},\:\mathrm{c},\:\mathrm{d}\:\in\:\mathbb{R} \\ $$

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=∓2 then LHS− a^5 +b^5 +c^5 +d^4 (∓2)^5 +0^2 +0^2 +0^2 ∓32 RHS

$$ \\ $$$$\mathrm{ans} \\ $$$$ \\ $$$$\mathrm{a}^{\mathrm{4}} +\mathrm{b}^{\mathrm{4}} +\mathrm{c}^{\mathrm{4}} +\mathrm{d}^{\mathrm{4}} =\mathrm{16} \\ $$$$\mathrm{puting}\:\:\mathrm{b}=\mathrm{c}=\mathrm{d}=\mathrm{0} \\ $$$$\mathrm{a}^{\mathrm{4}} +\mathrm{0}^{\mathrm{2}} +\mathrm{0}^{\mathrm{2}} +\mathrm{0}^{\mathrm{2}} =\mathrm{16} \\ $$$$\mathrm{a}^{\mathrm{4}} =\mathrm{16} \\ $$$$\mathrm{a}^{\mathrm{4}} =\mathrm{2}^{\mathrm{4}} \\ $$$$\mathrm{a}=\mp\mathrm{2} \\ $$$$\mathrm{then} \\ $$$$ \\ $$$$\mathrm{LHS}− \\ $$$$\mathrm{a}^{\mathrm{5}} +\mathrm{b}^{\mathrm{5}} +\mathrm{c}^{\mathrm{5}} +\mathrm{d}^{\mathrm{4}} \\ $$$$\left(\mp\mathrm{2}\right)^{\mathrm{5}} +\mathrm{0}^{\mathrm{2}} +\mathrm{0}^{\mathrm{2}} +\mathrm{0}^{\mathrm{2}} \\ $$$$\mp\mathrm{32}\:\:\:\mathrm{RHS} \\ $$$$ \\ $$

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.

$$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{any}\:\mathrm{4} \\ $$$${a},{b},{c}\:\mathrm{and}\:{d}\:\mathrm{sum}\:\mathrm{will}\:\mathrm{remain}\:\mathrm{less} \\ $$$$\mathrm{that}\:\mathrm{32}.\:\mathrm{I}\:\mathrm{mean}\:\mathrm{the}\:\mathrm{general}\:\mathrm{case}. \\ $$

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

Σa^4 =16 ⇒ a^4 ≤ 16 ⇒ a−1 ≤ 1 Σa^5 = Σ(a−1)a^4 +Σa^4 = (≤ 16)+16 ≤ 32 .

$$\:\Sigma{a}^{\mathrm{4}} =\mathrm{16}\:\:\:\:\Rightarrow\:\:\:{a}^{\mathrm{4}} \leqslant\:\mathrm{16} \\ $$$$\Rightarrow\:\:{a}−\mathrm{1}\:\leqslant\:\mathrm{1} \\ $$$$\Sigma{a}^{\mathrm{5}} =\:\Sigma\left({a}−\mathrm{1}\right){a}^{\mathrm{4}} +\Sigma{a}^{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:=\:\left(\leqslant\:\mathrm{16}\right)+\mathrm{16}\:\leqslant\:\mathrm{32}\:. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com