Test 9

Pure Mathematics – Curve Sketching

p.1

King’s College 2007 – 2008 F.6 Pure Mathematics Revision Test 9 Time Allowed: 50 minutes Total Mark: 30 1.

(1998) 1 3

Let f ( x ) = x ( x + 1) 3 . (a) (i)

2

Find f’(x) for x ≠ −1, 0 .

(ii) Show that f ' ' (x ) =

−2 5 3

9 x (x + 1)

for x ≠ −1, 0 . 4 3

(2 marks) (b) Determine with reasons whether f’(-1) and f’(0) exist or not. (2 marks) (c) Determine the values of x for each of the following cases: (i) f’(x) > 0 (ii) f’(x) < 0 (iii) f’’(x) > 0 (iv) f’’(x) < 0 (3 marks) (d) Find all relative extrema and points of inflection of f(x). (2 marks) (e) Find all asymptotes of the graph of f(x). (2 marks) (f) Sketch the graph of f(x). (3 marks) 2.

(2008 - Modified) Let f : ℜ → ℜ be defined by f(x) = (2x2 – 14x + 25)e2x. (a) Find f’(x) and f’’(x). (4 marks) (b) Solve each of the following inequalities: (i) f(x) > 0 (ii) f’(x) > 0 (iii) f’’(x) > 0 (3 marks) (c) Find the point(s) of inflexion of the graph of y = f(x). (2 marks) (d) Find the asymptote(s) of the graph of y = f(x). (3 marks) (e) Sketch the graph of y = f(x). (3 marks) --End of Test--