Let $f : \mathbb{R} \rightarrow \mathbb{R}$. For each statement, state if it is true or false.
(a) There is no continuous function $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx < \frac{1}{4}$.
(b) There is only one continuous function $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx = \frac{1}{4}$.
(c) There are infinitely many continuous functions $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx = \frac{1}{4}$.