Number of continuous function(s)

Q&ACategory: Exam HelpNumber of continuous function(s)
AY Sir Staff asked 1 year ago

Number of continuous function(s) f: [0, 1] → R satisfying \int ^{1}_{0}f\left( x\right) dx=\dfrac{1}{3}+\int ^{1}_{0}f^{2}\left( x^{2}\right) dx is/are…

1 Answers
AY Sir Staff answered 1 year ago

Given that \begin{aligned}\int ^{1}_{0}f\left( x\right) dx=\dfrac{1}{3}+\int ^{1}_{0}f^{2}\left( x^{2}\right) dx\\<br /> \Rightarrow \dfrac{1}{3}=\int ^{1}_{0}x^{2}dx,\int ^{1}_{0}f\left( x\right) dx=\int ^{1}_{0}2xf\left( x^{2}\right) dx\\<br /> \Rightarrow \int ^{1}_{0}\left( f^{2}\left( x^{2}\right) +x^{2}-2xf\left( x^{2}\right) \right) dx=0\\<br /> \Rightarrow \int ^{1}_{0}\left( f\left( x^{2}\right) -x\right) dx=0\Rightarrow f\left( x^{2}\right) =x\\<br /> \Rightarrow f\left( x\right) =\sqrt{x}\end{aligned}
Thus the total number of such continuous function(s) is one.