Question

Number of positive continuous functions $f\left(x\right)$ defined in $[0,1]$ for which ${\int }_0^{1}f\left(x\right)dx=1$, ${\int }_0^{1}xf\left(x\right)dx=2$, ${\int }_0^1{x}^{2}f\left(x\right)dx=4$, is

• A. $1$
• B. $4$
• C. Infinite
• D. None of these

Answer 1

The correct option is D None of these

Using Integration by parts.
${\int }_0^1{x}^{2}f\left(x\right)dx={[x^2\int f\left(x\right)dx]}_0^1-{\int }_0^{1}2x\int f\left(x\right)dx$
Applying limits of integration, gives
${\int }_0^1{x}^{2}f\left(x\right)dx=1-4=-3$
But given that ${\int }_0^1{x}^{2}f\left(x\right)dx=4$ .
Hence , there is no continuous function $f\left(x\right)$ satisfying the given conditions.
ie, number of functions $=0$ ....Ans