Question and Answers Forum

All Questions      Topic List

Arithmetic Questions

Previous in All Question      Next in All Question      

Previous in Arithmetic      Next in Arithmetic      

Question Number 141405 by iloveisrael last updated on 18/May/21

Find the minimum value of k such that for arbitrary a,b >0 we have (a)^(1/(3 )) + (b)^(1/(3 )) ≤ k ((a+b))^(1/(3 ))

$$\:\:\:\:\:{Find}\:{the}\:{minimum}\:{value}\:{of}\:{k} \\ $$ $$\:\:\:\:\:{such}\:{that}\:{for}\:{arbitrary}\:{a},{b}\:>\mathrm{0} \\ $$ $$\:\:\:\:\:{we}\:{have}\:\:\sqrt[{\mathrm{3}\:}]{{a}}\:+\:\sqrt[{\mathrm{3}\:}]{{b}}\:\leqslant\:{k}\:\sqrt[{\mathrm{3}\:}]{{a}+{b}}\: \\ $$

Answered by EDWIN88 last updated on 18/May/21

consider the function f(x) = (x)^(1/(3 )) we have f ′(x)= (1/3) x^(−(2/3)) & f ′′(x)=−(2/9)x^(−(5/3)) for any x ∈ (0,∞) thus f(x) concave on the interval (0,∞). By Jensen′s inequality we deduce (1/2)f(a) + (1/2)f(b) ≤ f (((a+b)/2)) (((a)^(1/(3 )) + (b)^(1/(3 )) )/2) ≤ (((a+b)/2))^(1/(3 )) (a)^(1/(3 )) + (b)^(1/(3 )) ≤ ((2 ((a+b))^(1/(3 )) )/( (2)^(1/(3 )) )) = ((2 (4)^(1/3) )/2)(a+b) minimum value of k is (4)^(1/(3 )) ⋇

$$\:\mathrm{consider}\:\mathrm{the}\:\mathrm{function}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\sqrt[{\mathrm{3}\:}]{\mathrm{x}} \\ $$ $$\:\mathrm{we}\:\mathrm{have}\:\mathrm{f}\:'\left(\mathrm{x}\right)=\:\frac{\mathrm{1}}{\mathrm{3}}\:\mathrm{x}^{−\frac{\mathrm{2}}{\mathrm{3}}} \:\&\:\mathrm{f}\:''\left(\mathrm{x}\right)=−\frac{\mathrm{2}}{\mathrm{9}}\mathrm{x}^{−\frac{\mathrm{5}}{\mathrm{3}}} \\ $$ $$\:\mathrm{for}\:\mathrm{any}\:\mathrm{x}\:\in\:\left(\mathrm{0},\infty\right)\:\mathrm{thus}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{concave}\:\mathrm{on}\:\mathrm{the} \\ $$ $$\mathrm{interval}\:\left(\mathrm{0},\infty\right).\:\:\mathrm{By}\:\mathrm{Jensen}'\mathrm{s}\:\mathrm{inequality}\: \\ $$ $$\:\mathrm{we}\:\mathrm{deduce}\: \\ $$ $$\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{f}\left(\mathrm{a}\right)\:+\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{f}\left(\mathrm{b}\right)\:\leqslant\:\mathrm{f}\:\left(\frac{\mathrm{a}+\mathrm{b}}{\mathrm{2}}\right) \\ $$ $$\:\:\:\:\:\:\:\:\:\:\frac{\sqrt[{\mathrm{3}\:}]{\mathrm{a}}\:+\:\sqrt[{\mathrm{3}\:}]{\mathrm{b}}}{\mathrm{2}}\:\leqslant\:\sqrt[{\mathrm{3}\:}]{\frac{\mathrm{a}+\mathrm{b}}{\mathrm{2}}} \\ $$ $$\:\:\:\:\:\:\:\:\:\sqrt[{\mathrm{3}\:}]{\mathrm{a}}\:+\:\sqrt[{\mathrm{3}\:}]{\mathrm{b}}\:\leqslant\:\frac{\mathrm{2}\:\sqrt[{\mathrm{3}\:}]{\mathrm{a}+\mathrm{b}}}{\:\sqrt[{\mathrm{3}\:}]{\mathrm{2}}}\:=\:\frac{\mathrm{2}\:\sqrt[{\mathrm{3}}]{\mathrm{4}}}{\mathrm{2}}\left(\mathrm{a}+\mathrm{b}\right) \\ $$ $$\:\:\:\:\:\:\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:{k}\:\mathrm{is}\:\sqrt[{\mathrm{3}\:}]{\mathrm{4}}\:\divideontimes \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com