True/False Question :
Let $f:\left [ 0,1 \right ]\rightarrow \mathbb{R}$ be a continuous function such that $f\left ( x \right )\geq x^{3}$ for all $x \in \left [ 0,1 \right ]$ with $\int_{0}^{1}f\left ( x \right )dx=\frac{1}{4}$. Then $f\left ( x \right )=x^{3}$ for all $x \in \mathbb{R}$.