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Question Number 127500 by mr W last updated on 30/Dec/20

if sin x_1 +sin x_2 +...+sin x_(100) =0,  find the maximum value of  sin^5  x_1 +sin^5  x_2 +...+sin^5  x_(100) .  (x_1 ,x_2 ,...,x_(100) ∈R)

$${if}\:\mathrm{sin}\:{x}_{\mathrm{1}} +\mathrm{sin}\:{x}_{\mathrm{2}} +...+\mathrm{sin}\:{x}_{\mathrm{100}} =\mathrm{0}, \\ $$$${find}\:{the}\:{maximum}\:{value}\:{of} \\ $$$$\mathrm{sin}^{\mathrm{5}} \:{x}_{\mathrm{1}} +\mathrm{sin}^{\mathrm{5}} \:{x}_{\mathrm{2}} +...+\mathrm{sin}^{\mathrm{5}} \:{x}_{\mathrm{100}} . \\ $$$$\left({x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,...,{x}_{\mathrm{100}} \in\mathbb{R}\right) \\ $$

Answered by mr W last updated on 31/Dec/20

say n values of sin x_k  are 1, that  means the other 100−n values of  sin x_k  are −(n/(100−n)).  S=n×1^5 +(100−n)(−(n/(100−n)))^5 =n−(n^5 /((100−n)^4 ))  let n=x  (dS/dx)=1−((5x^4 )/((100−x)^4 ))−((4x^5 )/((100−x)^5 ))=0  ((x^4 (500−x))/((100−x)^5 ))=1  ⇒x≈37.73 ⇒n=38  S_(max) =38−(100−38)(((38)/(100−38)))^5 =((30 141 600)/(923 521))  ≈32.638

$${say}\:{n}\:{values}\:{of}\:\mathrm{sin}\:{x}_{{k}} \:{are}\:\mathrm{1},\:{that} \\ $$$${means}\:{the}\:{other}\:\mathrm{100}−{n}\:{values}\:{of} \\ $$$$\mathrm{sin}\:{x}_{{k}} \:{are}\:−\frac{{n}}{\mathrm{100}−{n}}. \\ $$$${S}={n}×\mathrm{1}^{\mathrm{5}} +\left(\mathrm{100}−{n}\right)\left(−\frac{{n}}{\mathrm{100}−{n}}\right)^{\mathrm{5}} ={n}−\frac{{n}^{\mathrm{5}} }{\left(\mathrm{100}−{n}\right)^{\mathrm{4}} } \\ $$$${let}\:{n}={x} \\ $$$$\frac{{dS}}{{dx}}=\mathrm{1}−\frac{\mathrm{5}{x}^{\mathrm{4}} }{\left(\mathrm{100}−{x}\right)^{\mathrm{4}} }−\frac{\mathrm{4}{x}^{\mathrm{5}} }{\left(\mathrm{100}−{x}\right)^{\mathrm{5}} }=\mathrm{0} \\ $$$$\frac{{x}^{\mathrm{4}} \left(\mathrm{500}−{x}\right)}{\left(\mathrm{100}−{x}\right)^{\mathrm{5}} }=\mathrm{1} \\ $$$$\Rightarrow{x}\approx\mathrm{37}.\mathrm{73}\:\Rightarrow{n}=\mathrm{38} \\ $$$${S}_{{max}} =\mathrm{38}−\left(\mathrm{100}−\mathrm{38}\right)\left(\frac{\mathrm{38}}{\mathrm{100}−\mathrm{38}}\right)^{\mathrm{5}} =\frac{\mathrm{30}\:\mathrm{141}\:\mathrm{600}}{\mathrm{923}\:\mathrm{521}} \\ $$$$\approx\mathrm{32}.\mathrm{638} \\ $$

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