• # question_answer The number of continuous function $f:[0,1]\to [0,1],$ such that $f(x)<{{x}^{2}}$ for all x and $\int_{0}^{1}{f(x)dx=\frac{1}{3}}$ is A) 0          B) 1 C) 2                                  D) infinite

[a] Given, $f(x)<{{x}^{2}}f(x)$ is always positive for $x\in [0,1]$ $\int_{0}^{1}{f(x)dx<\int_{0}^{1}{{{x}^{2}}dx}}$ $\frac{1}{3}<\left| \frac{{{x}^{3}}}{3} \right|_{0}^{1}$$\Rightarrow$$\frac{1}{3}<\frac{1}{3}$ It is not possible.