1.8: Past-paper question 11

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 constants,

(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 $\displaystyle f(x)$ has its greatest value

(ii) the greatest value of $\displaystyle f(x)$

The curve $\displaystyle C$ has equation $\displaystyle y = f(x)$

The curve $\displaystyle S$ with equation $\displaystyle y = x^2 - x + 13$ intersects curve $\displaystyle C$ at two points.

(c) Find the $\displaystyle x$ coordinate of each of these two points.

(d) Use algebraic integration to find the exact area of the finite region bounded by the curve $\displaystyle C$ and the curve $\displaystyle S$.