Question Number 3832 by Yozzii last updated on 21/Dec/15
$${Show}\:{that},\:\forall{a},{b}\in\mathbb{R}^{+} , \\ $$$$\:\left(\frac{{a}}{{b}}\right)^{\mathrm{1}/\mathrm{3}} +\left(\frac{{b}}{{a}}\right)^{\mathrm{1}/\mathrm{3}} \leqslant\left\{\mathrm{2}\left({a}+{b}\right)\left(\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}\right)\right\}^{\mathrm{1}/\mathrm{3}} . \\ $$$$ \\ $$
Commented by RasheedSindhi last updated on 22/Dec/15
$$\left(\frac{{a}}{{b}}\right)^{\mathrm{1}/\mathrm{3}} +\left(\frac{{b}}{{a}}\right)^{\mathrm{1}/\mathrm{3}} \leqslant\left\{\mathrm{2}\left({a}+{b}\right)\left(\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}\right)\right\}^{\mathrm{1}/\mathrm{3}} \\ $$$${RHS}:\left\{\mathrm{2}\left({a}+{b}\right)\left(\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}\right)\right\}^{\mathrm{1}/\mathrm{3}} \\ $$$$=\left\{\mathrm{2}\left({a}+{b}\right)\left(\frac{{a}+{b}}{{ab}}\right)\right\}^{\mathrm{1}/\mathrm{3}} \\ $$$$=\left\{\frac{\mathrm{2}\left({a}+{b}\right)^{\mathrm{2}} }{{ab}}\right\}^{\mathrm{1}/\mathrm{3}} =\frac{\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left({a}+{b}\right)^{\mathrm{2}/\mathrm{3}} }{\left({ab}\right)^{\mathrm{1}/\mathrm{3}} } \\ $$$${LHS}:\left(\frac{{a}}{{b}}\right)^{\mathrm{1}/\mathrm{3}} +\left(\frac{{b}}{{a}}\right)^{\mathrm{1}/\mathrm{3}} \\ $$$$=\frac{{a}^{\mathrm{1}/\mathrm{3}} }{{b}^{\mathrm{1}/\mathrm{3}} }+\frac{{b}^{\mathrm{1}/\mathrm{3}} }{{a}^{\mathrm{1}/\mathrm{3}} }=\frac{{a}^{\mathrm{2}/\mathrm{3}} +{b}^{\mathrm{2}/\mathrm{3}} }{\left({ab}\right)^{\mathrm{1}/\mathrm{3}} } \\ $$$${LHS}\leqslant{RHS} \\ $$$$\frac{{a}^{\mathrm{2}/\mathrm{3}} +{b}^{\mathrm{2}/\mathrm{3}} }{\left({ab}\right)^{\mathrm{1}/\mathrm{3}} }\leqslant\frac{\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left({a}+{b}\right)^{\mathrm{2}/\mathrm{3}} }{\left({ab}\right)^{\mathrm{1}/\mathrm{3}} } \\ $$$$\Rightarrow{a}^{\mathrm{2}/\mathrm{3}} +{b}^{\mathrm{2}/\mathrm{3}} \leqslant\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left({a}+{b}\right)^{\mathrm{2}/\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\because\:{a},{b}\in\mathbb{R}^{+} \right] \\ $$$$\Rightarrow{a}^{\mathrm{2}/\mathrm{3}} +{b}^{\mathrm{2}/\mathrm{3}} \leqslant\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}/\mathrm{3}} \left({a}+{b}\right)^{\mathrm{2}/\mathrm{3}} \\ $$$$\Rightarrow{a}^{\mathrm{2}/\mathrm{3}} +{b}^{\mathrm{2}/\mathrm{3}} \leqslant\left(\sqrt{\mathrm{2}}{a}+\sqrt{\mathrm{2}}{b}\right)^{\mathrm{2}/\mathrm{3}} \\ $$$${Case}\left({i}\right)\:{a}={b} \\ $$$${a}^{\mathrm{2}/\mathrm{3}} +{b}^{\mathrm{2}/\mathrm{3}} \leqslant\left(\sqrt{\mathrm{2}}{a}+\sqrt{\mathrm{2}}{b}\right)^{\mathrm{2}/\mathrm{3}} \\ $$$$\Rightarrow\mathrm{2}{a}^{\mathrm{2}/\mathrm{3}} \leqslant\left(\mathrm{2}\sqrt{\mathrm{2}}{a}\right)^{\mathrm{2}/\mathrm{3}} \\ $$$$\vdots \\ $$$${Case}\left({ii}\right)\:{a}>{b} \\ $$$${a}^{\mathrm{2}/\mathrm{3}} +{b}^{\mathrm{2}/\mathrm{3}} \leqslant\left(\sqrt{\mathrm{2}}{a}+\sqrt{\mathrm{2}}{b}\right)^{\mathrm{2}/\mathrm{3}} \\ $$$$\Rightarrow{a}^{\mathrm{2}/\mathrm{3}} \left(\mathrm{1}+\left(\frac{{b}}{{a}}\right)^{\mathrm{2}/\mathrm{3}} \right)\leqslant\left(\sqrt{\mathrm{2}}{a}\right)^{\mathrm{2}/\mathrm{3}} \left\{\mathrm{1}+\frac{{b}}{{a}}\right\}^{\mathrm{2}/\mathrm{3}} \\ $$$$\Rightarrow{a}^{\mathrm{2}/\mathrm{3}} \left(\mathrm{1}+\left(\frac{{b}}{{a}}\right)^{\mathrm{2}/\mathrm{3}} \right)\leqslant\mathrm{2}^{\mathrm{1}/\mathrm{3}} {a}^{\mathrm{2}/\mathrm{3}} \left\{\mathrm{1}+\frac{{b}}{{a}}\right\}^{\mathrm{2}/\mathrm{3}} \\ $$$${Dividing}\:{by}\:{a}^{\mathrm{2}/\mathrm{3}} \left(>\mathrm{0}\right) \\ $$$$\Rightarrow\left(\mathrm{1}+\left(\frac{{b}}{{a}}\right)^{\mathrm{2}/\mathrm{3}} \right)\leqslant\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left\{\mathrm{1}+\frac{{b}}{{a}}\right\}^{\mathrm{2}/\mathrm{3}} \\ $$$${Using}\:{binomial}\:{theorm}\: \\ $$$$\left(\mathrm{1}+{x}\right)^{{n}} =\mathrm{1}+{nx}+\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}!}{x}^{\mathrm{2}} +… \\ $$$${where}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left(\mathrm{1}+\frac{{b}}{{a}}\right)^{\mathrm{2}/\mathrm{3}} =\mathrm{1}+\left(\frac{\mathrm{2}}{\mathrm{3}}\right)\left(\frac{{b}}{{a}}\right)^{\mathrm{2}/\mathrm{3}} +… \\ $$$${Continue} \\ $$
Commented by Yozzii last updated on 22/Dec/15
$${There}'{s}\:{an}\:{error}\:{in}\:{the}\:{third}\:{line}\: \\ $$$${of}\:{your}\:{attempt}.\: \\ $$$$'…=\left\{\frac{\mathrm{2}\left({a}+{b}\right)^{\mathrm{2}} }{{ab}}\right\}^{\mathrm{1}/\mathrm{3}} =\frac{\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left({a}+{b}\right)^{\mathrm{2}/\mathrm{3}} }{\left({ab}\right)^{\mathrm{1}/\mathrm{3}} }…' \\ $$$$ \\ $$
Commented by Rasheed Soomro last updated on 22/Dec/15
$$\mathcal{T}{han}\Bbbk\mathcal{S}!\:{I}\:{am}\:{going}\:{to}\:{correct}\:{it}. \\ $$
Answered by Rasheed Soomro last updated on 23/Dec/15
$${Show}\:{that},\:\forall{a},{b}\in\mathbb{R}^{+} , \\ $$$$\:\left(\frac{{a}}{{b}}\right)^{\mathrm{1}/\mathrm{3}} +\left(\frac{{b}}{{a}}\right)^{\mathrm{1}/\mathrm{3}} \leqslant\left\{\mathrm{2}\left({a}+{b}\right)\left(\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}\right)\right\}^{\mathrm{1}/\mathrm{3}} ………………\left(\boldsymbol{\mathrm{I}}\right) \\ $$$$−−−−−−−−−−−−−−− \\ $$$${A}\leqslant{B}\:\Leftrightarrow{A}^{\mathrm{3}} \leqslant{B}^{\mathrm{3}} \:\:,\:\:\:\forall{A},{B}>\mathrm{0} \\ $$$$\left(\frac{{a}}{{b}}\right)^{\mathrm{1}/\mathrm{3}} +\left(\frac{{b}}{{a}}\right)^{\mathrm{1}/\mathrm{3}} \leqslant\left\{\mathrm{2}\left({a}+{b}\right)\left(\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}\right)\right\}^{\mathrm{1}/\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\Rightarrow\:\left[\left(\frac{{a}}{{b}}\right)^{\mathrm{1}/\mathrm{3}} +\left(\frac{{b}}{{a}}\right)^{\mathrm{1}/\mathrm{3}} \right]^{\mathrm{3}} \leqslant\left[\left\{\mathrm{2}\left({a}+{b}\right)\left(\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}\right)\right\}^{\mathrm{1}/\mathrm{3}} \right]^{\mathrm{3}} \\ $$$$\Rightarrow\frac{{a}}{{b}}+\mathrm{3}\left(\frac{{a}}{{b}}\right)^{\mathrm{1}/\mathrm{3}} \left(\frac{{b}}{{a}}\right)^{\mathrm{1}/\mathrm{3}} \left\{\left(\frac{{a}}{{b}}\right)^{\mathrm{1}/\mathrm{3}} +\left(\frac{{b}}{{a}}\right)^{\mathrm{1}/\mathrm{3}} \right\}+\frac{{b}}{{a}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\leqslant\mathrm{2}\left({a}+{b}\right)\left(\frac{{a}+{b}}{{ab}}\right) \\ $$$$\Rightarrow\frac{{a}}{{b}}+\mathrm{3}\left(\left(\frac{{a}}{{b}}\right)^{\mathrm{1}/\mathrm{3}} +\left(\frac{{b}}{{a}}\right)^{\mathrm{1}/\mathrm{3}} \right)+\frac{{b}}{{a}}\leqslant\frac{\mathrm{2}\left({a}+{b}\right)^{\mathrm{2}} }{{ab}} \\ $$$$\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }{{ab}}+\mathrm{3}\left(\frac{{a}^{\mathrm{2}/\mathrm{3}} +{b}^{\mathrm{2}/\mathrm{3}} }{{a}^{\mathrm{1}/\mathrm{3}} {b}^{\mathrm{1}/\mathrm{3}} }\right)\leqslant\mathrm{2}\left(\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }{{ab}}\right)+\frac{\mathrm{4}{ab}}{{ab}} \\ $$$$\mathrm{3}\left(\frac{{a}^{\mathrm{2}/\mathrm{3}} +{b}^{\mathrm{2}/\mathrm{3}} }{{a}^{\mathrm{1}/\mathrm{3}} {b}^{\mathrm{1}/\mathrm{3}} }\right)\leqslant\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }{{ab}}+\mathrm{4} \\ $$$$\mathrm{3}\left(\frac{{a}^{\mathrm{2}/\mathrm{3}} +{b}^{\mathrm{2}/\mathrm{3}} }{{a}^{\mathrm{1}/\mathrm{3}} {b}^{\mathrm{1}/\mathrm{3}} }\right){ab}\leqslant{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +\mathrm{4}{ab}\:\:\left[{Multiplying}\:{by}\:{ab}>\mathrm{0}\right] \\ $$$$\mathrm{3}\left({a}^{\mathrm{2}/\mathrm{3}} +{b}^{\mathrm{2}/\mathrm{3}} \right){a}^{\mathrm{2}/\mathrm{3}} {b}^{\mathrm{2}/\mathrm{3}} \leqslant{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +\mathrm{4}{ab} \\ $$$$ \\ $$$${Continue} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\: \\ $$
Commented by Yozzii last updated on 24/Dec/15
$${Thanks}\:{for}\:{trying}\:{this}\:{question}.\: \\ $$$${It}\:{was}\:{taken}\:{from}\:{a}\:{national}\:{math} \\ $$$${olympiad}\:{exam}\:\left({Czech}\:{and}\:{Slovak}\right. \\ $$$$\left.{Republic}\right)\:{in}\:{the}\:{year}\:\mathrm{2000}.\: \\ $$$${I}\:{haven}'{t}\:{gotten}\:{anywhere}\:{near}\:{starting} \\ $$$${to}\:{solve}\:{the}\:{problem}.\: \\ $$
Commented by Rasheed Soomro last updated on 24/Dec/15
$$\mathcal{M}{y}\:\mathcal{E}{nglish}\:{is}\:{poor}.{So}\:\mathcal{I} \\ $$$$\mathcal{C}{ouldn}'{t}\:{understand}\:{sense}\:{of}\:{last} \\ $$$${two}\:\:{lines}.{Pl}\:{explain}. \\ $$
Commented by Yozzii last updated on 24/Dec/15
$${The}\:{problem}\:{is}\:{so}\:{difficult}\:{for}\:{me} \\ $$$${that},\:{it}\:{is}\:{taking}\:{a}\:{lot}\:{of}\:{time}\:{for}\:{me} \\ $$$${to}\:{solve}\:{it}.\: \\ $$
Commented by Rasheed Soomro last updated on 24/Dec/15
$$\mathbb{T}\mathrm{han}\Bbbk\mathbb{S}! \\ $$
Answered by Rasheed Soomro last updated on 23/Dec/15
$$\Rightarrow\mathrm{1}+\left(\frac{{b}}{{a}}\right)^{\mathrm{2}/\mathrm{3}} \leqslant\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left(\mathrm{1}+\frac{{b}}{{a}}\right)^{\mathrm{2}/\mathrm{3}} \:\:\:\:\left[\because{a}^{\mathrm{2}/\mathrm{3}} >\mathrm{0},{dividing}\:{by}\:{it}\right] \\ $$$$\:\left(\frac{{a}}{{b}}\right)^{\mathrm{1}/\mathrm{3}} +\left(\frac{{b}}{{a}}\right)^{\mathrm{1}/\mathrm{3}} \leqslant\left\{\mathrm{2}\left({a}+{b}\right)\left(\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}\right)\right\}^{\mathrm{1}/\mathrm{3}} \\ $$$$\frac{{a}^{\mathrm{1}/\mathrm{3}} }{{b}^{\mathrm{1}/\mathrm{3}} }+\frac{{b}^{\mathrm{1}/\mathrm{3}} }{{a}^{\mathrm{1}/\mathrm{3}} }\leqslant\left\{\mathrm{2}\left({a}+{b}\right)\left(\frac{{a}+{b}}{{ab}}\right)\right\}^{\mathrm{1}/\mathrm{3}} \\ $$$$\frac{{a}^{\mathrm{2}/\mathrm{3}} +{b}^{\mathrm{2}/\mathrm{3}} }{\left({ab}\right)^{\mathrm{1}/\mathrm{3}} }\leqslant\left\{\frac{\mathrm{2}\left({a}+{b}\right)^{\mathrm{2}} }{{ab}}\right\}^{\mathrm{1}/\mathrm{3}} \\ $$$$\frac{{a}^{\mathrm{2}/\mathrm{3}} +{b}^{\mathrm{2}/\mathrm{3}} }{\left({ab}\right)^{\mathrm{1}/\mathrm{3}} }\leqslant\frac{\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left({a}+{b}\right)^{\mathrm{2}/\mathrm{3}} }{\left({ab}\right)^{\mathrm{1}/\mathrm{3}} } \\ $$$${a}^{\mathrm{2}/\mathrm{3}} +{b}^{\mathrm{2}/\mathrm{3}} \:\leqslant\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left({a}+{b}\right)^{\mathrm{2}/\mathrm{3}} \left({i}\right)\:\:\:\left[\because\left({ab}\right)^{\mathrm{1}/\mathrm{3}} >\mathrm{0}\:,{so}\:{by}\:{multiplying}\right] \\ $$$${Case}\left({I}\right)\:{a}={b} \\ $$$$\left({i}\right)\:{will}\:{be}\: \\ $$$$\mathrm{2}{a}^{\mathrm{2}/\mathrm{3}} \leqslant\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left(\mathrm{2}{a}\right)^{\mathrm{2}/\mathrm{3}} \\ $$$$\mathrm{2}{a}^{\mathrm{2}/\mathrm{3}} \leqslant\mathrm{2}^{\mathrm{1}/\mathrm{3}} \mathrm{2}^{\mathrm{2}/\mathrm{3}} {a}^{\mathrm{2}/\mathrm{3}} \\ $$$$\mathrm{2}{a}^{\mathrm{2}/\mathrm{3}} \leqslant\mathrm{2}{a}^{\mathrm{2}/\mathrm{3}} \\ $$$$\therefore\:\:\leqslant\:{holds}\: \\ $$$${Case}\left({II}\right)\:{a}>{b} \\ $$$$\left({i}\right)\:\:{will}\:{be} \\ $$$${a}^{\mathrm{2}/\mathrm{3}} \left(\mathrm{1}+\left(\frac{{b}}{{a}}\right)^{\mathrm{2}/\mathrm{3}} \right)\leqslant\mathrm{2}^{\mathrm{1}/\mathrm{3}} {a}^{\mathrm{2}/\mathrm{3}} \left(\mathrm{1}+\frac{{b}}{{a}}\right)^{\mathrm{2}/\mathrm{3}} \\ $$$$\Rightarrow\mathrm{1}+\left(\frac{{b}}{{a}}\right)^{\mathrm{2}/\mathrm{3}} \leqslant\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left(\mathrm{1}+\frac{{b}}{{a}}\right)^{\mathrm{2}/\mathrm{3}} \:\:\:\:\left[\because{a}^{\mathrm{2}/\mathrm{3}} >\mathrm{0},{dividing}\:{by}\:{it}\right] \\ $$$$\Rightarrow\mathrm{1}+\left(\frac{{b}^{\mathrm{2}} }{{a}^{\mathrm{2}} }\right)^{\mathrm{1}/\mathrm{3}} \leqslant\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left\{\left(\mathrm{1}+\frac{{b}}{{a}}\right)^{\mathrm{2}} \right\}^{\mathrm{1}/\mathrm{3}} \\ $$$$\Rightarrow\mathrm{1}+\left(\frac{{b}^{\mathrm{2}} }{{a}^{\mathrm{2}} }\right)^{\mathrm{1}/\mathrm{3}} \leqslant\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left(\mathrm{1}+\frac{\mathrm{2}{b}}{{a}}+\frac{{b}^{\mathrm{2}} }{{a}^{\mathrm{2}} }\right)^{\mathrm{1}/\mathrm{3}} \\ $$$${Let}\:\frac{{b}}{{a}}={x}\:{then}\:{for}\:{x}<\mathrm{1} \\ $$$$\mathrm{1}+{x}^{\mathrm{2}/\mathrm{3}} \leqslant\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left(\mathrm{1}+\mathrm{2}{x}+{x}^{\mathrm{2}} \right)^{\mathrm{1}/\mathrm{3}} \\ $$$${Or}\:\mathrm{1}+{x}^{\mathrm{2}/\mathrm{3}} \leqslant\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left(\mathrm{1}+{x}\right)^{\mathrm{2}/\mathrm{3}} \\ $$$$\mathcal{I}{t}\:{couldn}'{t}\:{be}\:{solved}\:{by}\:{me}. \\ $$
Commented by RasheedSindhi last updated on 25/Dec/15
$${In}\:\mathrm{4}^{{th}} \:{line}\:{where}\:{does}\:\frac{\mathrm{2}}{\mathrm{3}}\:{disappear}? \\ $$
Commented by prakash jain last updated on 25/Dec/15
$$\mathrm{1}+{x}^{\mathrm{2}/\mathrm{3}} \leqslant\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left(\mathrm{1}+{x}\right)^{\mathrm{2}/\mathrm{3}} \\ $$$${f}\left({x}\right)=\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left(\mathrm{1}+{x}\right)^{\mathrm{2}/\mathrm{3}} −\mathrm{1}−{x}^{\mathrm{2}/\mathrm{3}} \\ $$$${f}\:'\left({x}\right)=\mathrm{2}^{\mathrm{1}/\mathrm{3}} \frac{\mathrm{2}}{\mathrm{3}}\left(\mathrm{1}+{x}\right)^{−\mathrm{1}/\mathrm{3}} −\frac{\mathrm{2}}{\mathrm{3}}{x}^{−\mathrm{1}/\mathrm{3}} \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}}\left(\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left(\mathrm{1}+{x}\right)^{−\mathrm{1}/\mathrm{3}} −{x}^{−\mathrm{1}/\mathrm{3}} \right) \\ $$$$\frac{\mathrm{2}}{\mathrm{1}+{x}}−\frac{\mathrm{1}}{{x}}=\frac{\mathrm{2}{x}−\mathrm{1}−{x}}{{x}\left(\mathrm{1}+{x}\right)}<\mathrm{0}\:\mathrm{for}\:\mathrm{0}<{x}<\mathrm{1} \\ $$$$\frac{\mathrm{2}}{\mathrm{1}+{x}}<\frac{\mathrm{1}}{{x}}\Rightarrow\mathrm{2}^{\mathrm{1}/\mathrm{3}} \left(\mathrm{1}+{x}\right)^{−\mathrm{1}/\mathrm{3}} <{x}^{−\mathrm{1}/\mathrm{3}} \\ $$$${f}'\left({x}\right)\leqslant\mathrm{0}\:\mathrm{0}<{x}<\mathrm{1} \\ $$$${f}\left({x}\right)\:\mathrm{is}\:\mathrm{decreasing}. \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{2}^{\mathrm{1}/\mathrm{3}} −\mathrm{1}\geqslant\mathrm{0} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{0} \\ $$$$\mathrm{Since}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{strictly}\:\mathrm{decreasing}\:\mathrm{for}\:\mathrm{0}<{x}<\mathrm{1} \\ $$$$\mathrm{and}\:{f}\left(\mathrm{0}\right)>\mathrm{0}\:\mathrm{and}\:{f}\left(\mathrm{1}\right)=\mathrm{0}{f}\left({x}\right)>\mathrm{0}\:{for}\:\mathrm{0}<{x}<\mathrm{1}. \\ $$$${b}>{a}\:{need}\:{not}\:{be}\:{checked}\:{since}\:{inequality} \\ $$$${is}\:{symmetric}\:{in}\:{a}\:{and}\:{b}. \\ $$$${x}={b}=\mathrm{0}\:\mathrm{LHS}\:\mathrm{and}\:\mathrm{RHS}\:\mathrm{are}\:\mathrm{undefined}. \\ $$
Commented by prakash jain last updated on 25/Dec/15
$$\mathrm{We}\:\mathrm{are}\:\mathrm{only}\:\mathrm{interested}\:\mathrm{in}\:\mathrm{sign}. \\ $$
Commented by Rasheed Soomro last updated on 25/Dec/15
$$\mathbb{T}\mathrm{h}\alpha{n}\Bbbk\mathfrak{S}! \\ $$