2023 EJC P2 Q4
2023 EJC P2 Q4
9 marks
Prelims
A curve \(C\) has equation
\({{x}^{3}}+{{y}^{3}}-xy=A\)
where \(A\) is a non-zero constant.
- Show that any stationary points on \(C\) lie on\(y=3{{x}^{2}}\).[2]
- Find, in terms of \(A\), the \(x\)-coordinates of the stationary points, and hence determine the range of values of \(A\) for which \(C\) has two distinct stationary points.[4]
- Suppose that \(C\) has two stationary points. Determine the nature of each of the stationary points found in part (b).[3]
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Answer:(ii) \(A>-\frac{1}{27}\) (iii) When \({{x}^{3}}=\frac{1+\sqrt{1+27A}}{27}\), \(\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}=-\frac{6}{\sqrt{1+27A}}<0\), so this is a maximum point.
When \({{x}^{3}}=\frac{1-\sqrt{1+27A}}{27}\), \(\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}=\frac{6}{\sqrt{1+27A}}>0\), so this is a minimum point.
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