19M.1.AHL.TZ1.H_7

pestleMathematicsAIHLPaper 119M· sl-5-3-differentiating-polynomials-n-e-zsource ↗

Find the coordinates of the points on the curve ${y^3} + 3x{y^2} - {x^3} = 27$ at which $\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0$ .

Markscheme / solution

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

attempt at implicit differentiation M1

$3{y^2}\frac{{{\text{d}}y}}{{{\text{d}}x}} + 3{y^2} + 6xy\frac{{{\text{d}}y}}{{{\text{d}}x}} - 3{x^2} = 0$ A1A1

Note: Award A1 for the second & third terms, A1 for the first term, fourth term & RHS equal to zero.

substitution of $\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0$ M1

$3{y^2} - 3{x^2} = 0$

$\Rightarrow y = \pm x$ A1

substitute either variable into original equation M1

$y = x \Rightarrow {x^3} = 9 \Rightarrow x = \sqrt[3]{9}$ (or ${y^3} = 9 \Rightarrow y = \sqrt[3]{9}$ ) A1

$y = - x \Rightarrow {x^3} = 27 \Rightarrow x = 3$ (or ${y^3} = - 27 \Rightarrow y = - 3$ ) A1

$\left( {\sqrt[3]{9}{\text{,}}\,\,\sqrt[3]{9}} \right)$ , (3, −3) A1

[9 marks]

Examiners’ report

[N/A]