TMUA 2020 · Paper 2 · Question 13

$f (x)$ is a function for which

\int_{0}^{3} (f (x))^{2} d x + \int_{0}^{3} f (x) d x = \int_{0}^{1} f (x) d x

Which of the following claims about $f (x)$ is/are necessarily true?

I $f (x) \leq 0$ for some $x$ with $1 \leq x \leq 3$

II $\int_{0}^{3} f (x) d x \leq \int_{0}^{1} f (x) d x$