Find the value of $p$.

Write down the coordinates of A.

Write down the rate of change of $f$ at A.

Find the coordinates of B.

Find the the rate of change of $f$ at B.

Let $R$ be the region enclosed by the graph of $f$ , the $x$-axis, the line $x = b$ and the line $x = a$. The region $R$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

evidence of valid approach     (M1)

eg$\,\,\,\,\,$$f(x) = 0,{\text{ }}y = 0$

2.73205

$p = 2.73$     A1     N2

[2 marks]

1.87938, 8.11721

$(1.88,{\text{ }}8.12)$     A2     N2

[2 marks]

rate of change is 0 (do not accept decimals)     A1     N1

[1 marks]

METHOD 1 (using GDC)

valid approach     M1

eg$\,\,\,\,\,$$f’’ = 0$, max/min on $f’,{\text{ }}x =  - 1$

sketch of either $f’$ or $f’’$, with max/min or root (respectively)     (A1)

$x = 1$     A1     N1

Substituting their $x$ value into $f$     (M1)

eg$\,\,\,\,\,$$f(1)$

$y = 4.5$     A1     N1

METHOD 2 (analytical)

$f’’ =  - 6{x^2} + 6$     A1

setting $f’’ = 0$     (M1)

$x = 1$     A1     N1

substituting their $x$ value into $f$     (M1)

eg$\,\,\,\,\,$$f(1)$

$y = 4.5$     A1     N1

[4 marks]

recognizing rate of change is $f’$     (M1)

eg$\,\,\,\,\,$$y’,{\text{ }}f’(1)$

rate of change is 6     A1     N2

[3 marks]

attempt to substitute either limits or the function into formula     (M1)

involving ${f^2}$ (accept absence of $\pi$ and/or ${\text{d}}x$)

eg$\,\,\,\,\,$$\pi \int {{{( - 0.5{x^4} + 3{x^2} + 2x)}^2}{\text{d}}x,{\text{ }}\int_1^{1.88} {{f^2}} }$

128.890

${\text{volume}} = 129$     A2     N3

[3 marks]