Let $p$ and $q$ be real numbers, and let $\displaystyle f(x) = (x^2 + px + q)\,\left|x^2 - 5x + 6\right| \quad \text{for every real } x.$ The first factor is an ordinary quadratic, and the second is the modulus of a quadratic with two real roots, so at first sight $f$ looks as if it has two corners. The numbers $p$ and $q$ are chosen so that $f$ is differentiable at every real number, with no exception at all. Here $p$ and $q$ are constants that do not depend on $x$, and the modulus is the ordinary absolute value of a real number. With these values of $p$ and $q$, what is the value of $f(5)$?