1.27: Completing the Square and Enclosed Area

4PM1/1R/June/2019 — Question 10 · 13 marks

f(x)=6x−x2,x∈R.f(x)=6x-x^2,\qquad x\in\mathbb{R}.
Given that f(x)\displaystyle f(x) can be written in the form D(x+E)2+F\displaystyle D(x+E)^2+F, where D\displaystyle D, E\displaystyle E and F\displaystyle F are integers,
(a) find the value of D\displaystyle D, the value of E\displaystyle E and the value of F\displaystyle F.
(3)
(b) Find
(i) the maximum value of f(x)\displaystyle f(x),
(ii) the value of x\displaystyle x for which the maximum occurs.
(2)
The curve C\displaystyle C has equation y=f(x)\displaystyle y=f(x).
The curve S\displaystyle S has equation
y=x2−4x+8.y=x^2-4x+8.
The curve S\displaystyle S intersects the curve C\displaystyle C at two points.
(c) Find the coordinates of each of these two points.
(4)
The finite region R\displaystyle R is bounded by the curve C\displaystyle C and the curve S\displaystyle S.
(d) Use algebraic integration to find the area of R\displaystyle R.
(4)