C3 January 2007 Q7

7. \[\mathrm{f}(x) = x^4 - 4x - 8.\]

(a) Show that there is a root of \(\mathrm{f}(x) = 0\) in the interval \([-2, -1]\). (3)

(b) Find the coordinates of the turning point on the graph of \(y = \mathrm{f}(x)\). (3)

(c) Given that \(\mathrm{f}(x) = (x - 2)(x^3 + ax^2 + bx + c)\), find the values of the constants, \(a\), \(b\) and \(c\). (3)

(d) In the space provided, sketch the graph of \(y = \mathrm{f}(x)\). (3)

(e) Hence sketch the graph of \(y = |\mathrm{f}(x)|\). (1)

Mark scheme (a)

Scheme	Marks
\(\mathrm{f}(-2) = 16 + 8 - 8\ (= 16) \gt 0\)	B1
\(\mathrm{f}(-1) = 1 + 4 - 8\ (= -3) \lt 0\)	B1
Change of sign (and continuity) \(\Rightarrow\) root in interval \((-2, -1)\) ft their calculation as long as there is a sign change	B1ft
	(3)

Scheme	Marks
\(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 4x^3 - 4 = 0 \Rightarrow x = 1\)	M1 A1
Turning point is \((1, -11)\)	A1
	(3)

Scheme	Marks
\(a = 2,\ b = 4,\ c = 4\)	B1 B1 B1
	(3)