1.21: Past-paper question 2

Given that $\displaystyle f(x)$ can be written in the form $\displaystyle A(x + B)^2 + C$, where $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$ are integers,

(a) find the value of $\displaystyle A$, the value of $\displaystyle B$ and the value of $\displaystyle C$

(b) Hence, or otherwise, find

(i) the value of $\displaystyle x$ for which $\displaystyle \frac{1}{f(x)}$ is a maximum

(ii) the maximum value of $\displaystyle \frac{1}{f(x)}$