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For-f-x-ax-n-b-when-f-o-f-is-continuous-Does-there-exist-a-solution-S-f-x-dx-lt-

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Question Number 4116 by Filup last updated on 29/Dec/15
For: f(x)=∣ax^n +b∣ when f(α) o f(β) is continuous, Does there exist a solution: S=∫_α ^( β) f(x)dx α<β
$$\mathrm{For}:\:{f}\left({x}\right)=\mid{ax}^{{n}} +{b}\mid \\ $$$$\mathrm{when}\:{f}\left(\alpha\right)\:\mathrm{o}\:{f}\left(\beta\right)\:\mathrm{is}\:\mathrm{continuous}, \\ $$$$\mathrm{Does}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{a}\:\mathrm{solution}: \\ $$$${S}=\int_{\alpha} ^{\:\beta} {f}\left({x}\right){dx} \\ $$$$\alpha<\beta \\ $$
Commented by Yozzii last updated on 29/Dec/15
f: R→R and α<β (α,β∈R) with f being defined at x=α,x=β. f(x)=∣ax^n +b∣ represents the modulus of a real polynomial,if n∈Z^+ +{0} and a,b∈R. Thus, f(x) is continuous ∀x∈R. ∴ f(x)=∣ax^n +b∣≥0⇒∫_α ^β f(x)dx≥0 ⇒∃S∈R^+ +{0} such that S=∫_α ^β f(x)dx. S≮0 since f(x)≮0 ∀x∈R. If n∈Z^− , and α<0<β then ∄S∈C such that S=∫_α ^β f(x)dx since the integral does not exist. ∫_α ^β f(x)dx=∫_α ^0 f(x)dx+∫_0 ^β f(x)dx =∫_α ^0 ∣(a/x^n )+b∣dx+∫_0 ^β ∣(a/x^n )+b∣dx f(0) is undefined so that each of the above integrals are undefined within the limit of x→0.
$${f}:\:\mathbb{R}\rightarrow\mathbb{R}\:{and}\:\alpha<\beta\:\:\:\left(\alpha,\beta\in\mathbb{R}\right) \\ $$$${with}\:{f}\:{being}\:{defined}\:{at}\:{x}=\alpha,{x}=\beta. \\ $$$${f}\left({x}\right)=\mid{ax}^{{n}} +{b}\mid\:{represents}\:{the}\:{modulus} \\ $$$${of}\:{a}\:{real}\:{polynomial},{if}\:{n}\in\mathbb{Z}^{+} +\left\{\mathrm{0}\right\} \\ $$$${and}\:{a},{b}\in\mathbb{R}. \\ $$$${Thus},\:{f}\left({x}\right)\:{is}\:{continuous}\:\forall{x}\in\mathbb{R}. \\ $$$$\therefore\:{f}\left({x}\right)=\mid{ax}^{{n}} +{b}\mid\geqslant\mathrm{0}\Rightarrow\int_{\alpha} ^{\beta} {f}\left({x}\right){dx}\geqslant\mathrm{0} \\ $$$$\Rightarrow\exists{S}\in\mathbb{R}^{+} +\left\{\mathrm{0}\right\}\:{such}\:{that}\:{S}=\int_{\alpha} ^{\beta} {f}\left({x}\right){dx}. \\ $$$${S}\nless\mathrm{0}\:{since}\:{f}\left({x}\right)\nless\mathrm{0}\:\forall{x}\in\mathbb{R}. \\ $$$$ \\ $$$${If}\:{n}\in\mathbb{Z}^{−} ,\:{and}\:\alpha<\mathrm{0}<\beta\:{then}\:\nexists{S}\in\mathbb{C}\:{such} \\ $$$${that}\:{S}=\int_{\alpha} ^{\beta} {f}\left({x}\right){dx}\:{since}\:{the}\:{integral} \\ $$$${does}\:{not}\:{exist}.\: \\ $$$$\int_{\alpha} ^{\beta} {f}\left({x}\right){dx}=\int_{\alpha} ^{\mathrm{0}} {f}\left({x}\right){dx}+\int_{\mathrm{0}} ^{\beta} {f}\left({x}\right){dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\int_{\alpha} ^{\mathrm{0}} \mid\frac{{a}}{{x}^{{n}} }+{b}\mid{dx}+\int_{\mathrm{0}} ^{\beta} \mid\frac{{a}}{{x}^{{n}} }+{b}\mid{dx} \\ $$$${f}\left(\mathrm{0}\right)\:{is}\:{undefined}\:{so}\:{that}\:{each}\:{of}\:{the} \\ $$$${above}\:{integrals}\:{are}\:{undefined}\:{within} \\ $$$${the}\:{limit}\:{of}\:{x}\rightarrow\mathrm{0}. \\ $$

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