Find f (x).

The graph of f has a point of inflexion at x = p. Find p.

Find the values of x for which the graph of f is concave-down.

evidence of integration       (M1)

eg  \(\int {f'\left( x \right)} \)

correct integration (accept absence of C)       (A1)(A1)

eg  \({x^3} + \frac{{18}}{2}{x^2} + C,\,\,{x^3} + 9{x^2}\)

attempt to substitute x = −1 into their f = 0 (must have C)      M1

eg  \({\left( { - 1} \right)^3} + 9{\left( { - 1} \right)^2} + C = 0,\,\, - 1 + 9 + C = 0\)

Note: Award M0 if they substitute into original or differentiated function.

correct working       (A1)

eg  \(8 + C = 0,\,\,\,C =  - 8\)

\(f\left( x \right) = {x^3} + 9{x^2} - 8\)      A1 N5

[6 marks]

METHOD 1 (using 2^nd derivative)

recognizing that f" = 0 (seen anywhere)      M1

correct expression for f"      (A1)

eg   6x + 18, 6p + 18

correct working      (A1)

6p + 18 = 0

p = −3       A1 N3

METHOD 1 (using 1^st derivative)

recognizing the vertex of f′ is needed       (M2)

eg   \( - \frac{b}{{2a}}\) (must be clear this is for f′)

correct substitution      (A1)

eg   \(\frac{{ - 18}}{{2 \times 3}}\)

p = −3       A1 N3

[4 marks]

valid attempt to use f" (x) to determine concavity      (M1)

eg   f" (x) < 0, f" (−2), f" (−4),  6x + 18 ≤ 0 

correct working       (A1)

eg   6x + 18 < 0, f" (−2) = 6, f" (−4) = −6 

f concave down for x < −3 (do not accept x ≤ −3)       A1 N2

[3 marks]