19N.2.SL.TZ0.S_8

pestleMathematicsAASLPaper 219N· sl-5-8-testing-for-max-and-min-optimisation-points-of-inflexionsource ↗

Let $f\left( x \right) = {x^4} - 54{x^2} + 60x$ , for $- 1 \leqslant x \leqslant 6$ . The following diagram shows the graph of $f$ .

There are $x$ -intercepts at $x = 0$ and at $x = p$ . There is a maximum at point ${\text{A}}$ where $x = a$ , and a point of inflexion at point ${\text{B}}$ where $x = b$ .

Find the value of $p$ .

[2]

Write down the coordinates of ${\text{A}}$ .

[2]

b.i.

Find the equation of the tangent to the graph of $f$ at ${\text{A}}$ .

[2]

b.ii.

Find the coordinates of ${\text{B}}$ .

[5]

c.i.

Find the rate of change of $f$ at ${\text{B}}$ .

[2]

c.ii.

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

[3]

Markscheme / solution

evidence of valid approach (M1)

eg $f\left( x \right) = 0$ , $y = 0$

$1.13843$

$p = 1.14$ A1 N2

[2 marks]

$0.562134$ , $16.7641$

$\left( {0.562{\text{, }}16.8} \right)$ A2 N2

[2 marks]

b.i.

valid approach (M1)

eg tangent at maximum point is horizontal, $f' = 0$

$y = 16.8$ (must be an equation) A1 N2

[2 marks]

b.ii.

METHOD 1 (using GDC)

valid approach M1

eg $f'' = 0$ , max/min on ${f'}$ , $x = - 3$

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

$x = 3$ A1 N1

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

eg $f\left( 3 \right)$

$y = - 225$ (exact) (accept $\left( {3{\text{, }} - 225} \right)$ ) A1 N1

METHOD 2 (analytical)

$f'' = 12{x^2} - 108$ A1

valid approach (M1)

eg $f'' = 0$ , $x = \pm 3$

$x = 3$ A1 N1

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

eg $f\left( 3 \right)$

$y = - 225$ (exact) (accept $\left( {3{\text{, }} - 225} \right)$ ) A1 N1

[5 marks]

c.i.

recognizing rate of change is ${f'}$ (M1)

eg ${y'}$ , $f'\left( 3 \right)$

rate of change is $-156$ (exact) A1 N2

[2 marks]

c.ii.

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

eg $\int_{1.14}^3 {{f^2}}$ , $\pi \int {{{\left( {{x^4} - 54{x^2} + 60x} \right)}^2}} {\text{d}}x$ , $25\,752.0$

$80\,902.3$

volume $= 80\,900$ A2 N3

[3 marks]

Examiners’ report

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b.i.

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b.ii.

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c.i.

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c.ii.

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