17N.1.SL.TZ0.S_6

pestleMathematicsAASLPaper 117N· sl-5-8-testing-for-max-and-min-optimisation-points-of-inflexionsource ↗

Let $f(x) = 15 - {x^2}$ , for $x \in \mathbb{R}$ . The following diagram shows part of the graph of $f$ and the rectangle OABC, where A is on the negative $x$ -axis, B is on the graph of $f$ , and C is on the $y$ -axis.

N17/5/MATME/SP1/ENG/TZ0/06

Find the $x$ -coordinate of A that gives the maximum area of OABC.

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

attempt to find the area of OABC (M1)

eg $\,\,\,\,\,$ ${\text{OA}} \times {\text{OC, }}x \times f(x),{\text{ }}f(x) \times ( - x)$

correct expression for area in one variable (A1)

eg $\,\,\,\,\,$ ${\text{area}} = x(15 - {x^2}),{\text{ }}15x - {x^3},{\text{ }}{x^3} - 15x$

valid approach to find maximum area (seen anywhere) (M1)

eg $\,\,\,\,\,$ $A’(x) = 0$

correct derivative A1

eg $\,\,\,\,\,$ $15 - 3{x^2},{\text{ }}(15 - {x^2}) + x( - 2x) = 0,{\text{ }} - 15 + 3{x^2}$

correct working (A1)

eg $\,\,\,\,\,$ $15 = 3{x^2},{\text{ }}{x^2} = 5,{\text{ }}x = \sqrt 5$

$x = - \sqrt 5 {\text{ }}\left( {{\text{accept A}}\left( { - \sqrt 5 ,{\text{ }}0} \right)} \right)$ A2 N3

[7 marks]

Examiners’ report

[N/A]