To write $\displaystyle\inf\int f(x)\,dx$ without the words that came after that fails to convey what was said. It says "for $C^\infty$ functions that vanish at $0$ and [. . . . .]", and the expression inside the integral has "$u$" in it. In other words, it identifies a particular set. An infimum is an infimum of a set.
The infimum of a set with a smallest member is the smallest member. Thus the infimum of the set of all nonnegative numbers is $0$. The infimum of some sets with no smallest member exists. For example, there is no smallest positive number, and the infimum of the set of all positive numbers is $0$.
The infimum of a set is the greatest lower bound of the set. $0$ is a lower bound of the set of all positive numbers because $0$ is less than or equal to every positive number. No number bigger than $0$ is a lower bound of the set of all positive numbers. So $0$ is the greatest among all lower bounds.