Question

The number of continuous function $f:[0,1]\to [0,1]$ such that $f\left(x\right)<x^2$ for all $x$ and $\underset{0}{\overset{1}{\int }}f\left(x\right)dx=\frac{1}{3}$ is

• A. $0$
• B. $1$
• C. $2$
• D. infinite

Answer 1

The correct option is A $0$

$f\left(x\right)<x^2\Rightarrow \underset{0}{\overset{1}{\int }}f\left(x\right)dx<\underset{0}{\overset{1}{\int }}{x}^{2}dx\Rightarrow \underset{0}{\overset{1}{\int }}f\left(x\right)dx<\frac{1}{3}$
But is given that $\underset{0}{\overset{1}{\int }}f\left(x\right)dx=\frac{1}{3}$
So no possible function.