# mathematical-analysis-f-C-0-1-and-0-1-x-n-f-x-dx-1-n-2-n-N-prove-f-x-x-

Question Number 142893 by mnjuly1970 last updated on 06/Jun/21
$$\:\:\:\:\:\:\:\:\: \\$$$$\:\:\:\:\:\:\:\:\:\:\:…..{mathematical}\:…..{analysis}…… \\$$$$\:\:\:\:\:\:\:{f}\:\in\:{C}\:\left[\mathrm{0},\mathrm{1}\right]\:{and}\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{{n}} {f}\left({x}\right){dx}=\frac{\mathrm{1}}{{n}+\mathrm{2}}\:,\:{n}\in\mathbb{N} \\$$$$\:\:\:\:\:\:\:\:{prove}\:\:{f}\left({x}\right):={x}\:….. \\$$
Answered by mindispower last updated on 06/Jun/21
$$\frac{\mathrm{1}}{{n}+\mathrm{2}}=\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}+\mathrm{1}} {dx} \\$$$$\Rightarrow\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} \left({f}\left({x}\right)−{x}\right){dx}=\mathrm{0},\forall{n}\in\mathbb{N} \\$$$${since}\:{f}\left({x}\right)\in{C}\left[\mathrm{0},\mathrm{1}\right],\exists{p}_{{m}} \:{of}\:{polynomial}\:{such}\:{that} \\$$$${lagrange}\:{theorem} \\$$$$\mid{f}\left({x}\right)−{p}_{{m}} \mid\rightarrow\mathrm{0} \\$$$$\Rightarrow\forall{n}\in\mathbb{N}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} {p}_{{m}} \left({x}\right){dx}=\mathrm{0},{p}_{{m}} \in\mathbb{R}\left[{X}\right] \\$$$${p}_{{m}} \left({x}\right)=\underset{{k}=\mathrm{0}} {\overset{{l}} {\sum}}\int_{\mathrm{0}} ^{\mathrm{1}} {a}_{{k}} {x}^{{k}} .{x}^{{n}} {dx}=\mathrm{0}\Rightarrow \\$$$$\underset{{k}=\mathrm{0}} {\overset{{l}} {\sum}}\frac{{a}_{{k}} }{{k}+{n}+\mathrm{1}}=\mathrm{0}{we}\:{got}\:{infintie}\:{linear}\:{equation}\: \\$$$$\Rightarrow\left({a}_{{k}} \right)=\mathrm{0},\forall{k}\in\left[\mathrm{0},{l}\right] \\$$$$\Rightarrow{p}_{{m}} =\mathrm{0},\forall{m}\in\mathbb{N} \\$$$$\Rightarrow\mid{p}_{{m}} −{x}\mid\rightarrow\mathrm{0}\Rightarrow{p}_{{m}} \rightarrow{x} \\$$$$\mid{f}\left({x}\right)−{x}\mid=\mid{f}\left({x}\right)−{p}_{{m}} +{p}_{{m}} −{x}\mid\leqslant\mid{p}_{{m}} −{x}\mid_{\mathrm{0}} +\mid{f}\left({x}\right)−{p}_{{m}} \mid_{=\mathrm{0}} \\$$$$\Rightarrow{of}\left({x}\right)−{x}=\mathrm{0}\Rightarrow{f}\left({x}\right)={x} \\$$
Commented by mnjuly1970 last updated on 06/Jun/21
$$\:\:{bravo}\:..\:{very}\:{nice}\:{mr}\:{power}… \\$$
Commented by mindispower last updated on 06/Jun/21
$${pleasur}\:{i}\:{love}\:{maths}\:{but}\:{i}/{stopped}\:{befor} \\$$$${my}\:{graduation}\:{so}\:{sad} \\$$
Commented by Ar Brandon last updated on 17/Jun/21
Oh ! Dommage ! Qu'est-ce qui s'est passé ?