Write down the gradient of the curve of $f$ at P.

Find the equation of the normal to the curve of $f$ at P.

Determine the concavity of the graph of $f$ when $4 < x < 5$ and justify your answer.

$- 2$     A1     N1

[1 mark]

gradient of normal $= \frac{1}{2}$     (A1)

attempt to substitute their normal gradient and coordinates of P (in any order)     (M1)

eg$\,\,\,\,\,$$y - 4 = \frac{1}{2}(x - 3),{\text{ }}3 = \frac{1}{2}(4) + b,{\text{ }}b = 1$

$y - 3 = \frac{1}{2}(x - 4),{\text{ }}y = \frac{1}{2}x + 1,{\text{ }}x - 2y + 2 = 0$     A1     N3

[3 marks]

correct answer and valid reasoning     A2     N2

answer:     eg     graph of $f$ is concave up, concavity is positive (between $4 < x < 5$)

reason:     eg     slope of $f’$ is positive, $f’$ is increasing, $f’’ > 0$,

sign chart (must clearly be for $f’’$ and show A and B)

Note:     The reason given must refer to a specific function/graph. Referring to “the graph” or “it” is not sufficient.

[2 marks]