4400 Higher November 2006 Paper 3 Q14

EdexcelHigherOld spec9 marksDifferentiation

(a) For the equation \(y = 5000x - 625x^2\), find \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\). (2)

(b) Find the coordinates of the turning point on the graph of \(y = 5000x - 625x^2\). (3)

(c)

(i) State whether this turning point is a maximum or a minimum.

(ii) Give a reason for your answer. (2)

(d) A publisher has to set the price for a new book.
The profit, £\(y\), depends on the price of the book, £\(x\), where \[y = 5000x - 625x^2\]

(i) What price would you advise the publisher to set for the book?

(ii) Give a reason for your answer. (2)

Scheme	Marks
\(5000 - 1250x\)	B2
	(2)

Mark scheme (c)

Scheme	Marks
(i) Answer: max	B1
(ii) coeff of \(x^2 \lt 0\) or \(\dfrac{\mathrm{d}y}{\mathrm{d}x} \gt 0\) for \(x\) value \(\lt 4\) and \(\dfrac{\mathrm{d}y}{\mathrm{d}x} \lt 0\) for \(x\) value \(\gt 4\) or \(y \lt 10\,000\) for \(x\) value \(\lt 4\) and for \(x\) value \(\gt 4\) or \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = -1250 \lt 0\)	B1
	(2)

B1: independent

B1: CHECK: the printed scheme has \(\dfrac{\mathrm{d}y}{\mathrm{d}x} \gt 0\) for \(x\) value \(\gt 4\); corrected here to \(\lt 0\)