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Question Number 155568 by mr W last updated on 02/Oct/21

$$fora,b,c,d,e\in Randa+b+c+d+e=5$$$$findtheminimumvalueof$$$$a^2+2b^2+3c^2+4d^2+5e^2=?$$

Answered by qaz last updated on 02/Oct/21

$${\mathrm{a}}^2+{2\mathrm{b}}^2+{3\mathrm{c}}^2+{4\mathrm{d}}^2+{5\mathrm{e}}^2$$$$={\mathrm{a}}^2+\frac{{\mathrm{b}}^2}{\frac{1}{2}}+\frac{{\mathrm{c}}^2}{\frac{1}{3}}+\frac{{\mathrm{c}}^2}{\frac{1}{4}}+\frac{{\mathrm{e}}^2}{\frac{1}{5}}$$$$⩾\frac{{(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}+\mathrm{e})}^2}{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}}$$$$=\frac{1500}{137}$$$$\mathrm{inequality}\mathrm{hold}\mathrm{only}\frac{\mathrm{a}}{1}=\frac{\mathrm{b}}{\frac{1}{2}}=\frac{\mathrm{c}}{\frac{1}{3}}=\frac{\mathrm{d}}{\frac{1}{4}}=\frac{\mathrm{e}}{\frac{1}{5}}$$$$\mathrm{ie}.\mathrm{a}=\frac{300}{137}\mathrm{b}=\frac{150}{137}\mathrm{c}=\frac{100}{137}\mathrm{d}=\frac{75}{137}\mathrm{e}=\frac{60}{137}$$

Commented by mr W last updated on 02/Oct/21

$$thankssir!$$

Answered by mr W last updated on 02/Oct/21

$$usingmethodasinQ155495:$$$$$$$$S=a^2+2b^2+3c^2+4d^2+5e^2$$$$=a^2+{\left(\frac{b}{\frac{1}{\sqrt{2}}}\right)}^2+{\left(\frac{c}{\frac{1}{\sqrt{3}}}\right)}^2+{\left(\frac{d}{\frac{1}{\sqrt{4}}}\right)}^2+{\left(\frac{e}{\frac{1}{\sqrt{5}}}\right)}^2$$$$={\left(a^{\prime }\right)}^2+{\left(b^{\prime }\right)}^2+{\left(c^{\prime }\right)}^2+{\left(d^{\prime }\right)}^2+{\left(e^{\prime }\right)}^2$$$$$$$$a+b+c+d+e=5$$$$a+\frac{1}{\sqrt{2}}\times \left(\frac{b}{\frac{1}{\sqrt{2}}}\right)+\frac{1}{\sqrt{3}}\times \left(\frac{c}{\frac{1}{\sqrt{3}}}\right)+\frac{1}{\sqrt{4}}\times \left(\frac{d}{\frac{1}{\sqrt{4}}}\right)+\frac{1}{\sqrt{5}}\times \left(\frac{e}{\frac{1}{\sqrt{5}}}\right)=5$$$$\Rightarrow a^{\prime }+\frac{b^{\prime }}{\sqrt{2}}+\frac{c^{\prime }}{\sqrt{3}}+\frac{d^{\prime }}{\sqrt{4}}+\frac{e^{\prime }}{\sqrt{5}}=5$$$$$$$$S_{min}={\left(\frac{5}{\sqrt{1^2+{\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{1}{\sqrt{3}}\right)}^2+{\left(\frac{1}{\sqrt{4}}\right)}^2+{\left(\frac{1}{\sqrt{5}}\right)}^2}}\right)}^2$$$$=\frac{25}{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}}=\frac{1500}{137}$$

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