Challenging +1.8 This AEA question requires optimization using Lagrange multipliers or substitution, finding critical points where x=±y, computing max/min values of x²+y² subject to x⁴+y⁴=1, sketching a quartic curve alongside a circle, and applying geometric insight about tangency. Multi-step with non-standard curve analysis, but the optimization is guided and techniques are accessible to strong A-level students.
6.(a)Given that \(x ^ { 4 } + y ^ { 4 } = 1\) ,prove that \(x ^ { 2 } + y ^ { 2 }\) is a maximum when \(x = \pm y\) ,and find the maximum and minimum values of \(x ^ { 2 } + y ^ { 2 }\) .
(b)On the same diagram,sketch the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations \(x ^ { 4 } + y ^ { 4 } = 1\) and \(x ^ { 2 } + y ^ { 2 } = 1\) respectively.
(c)Write down the equation of the circle \(C _ { 3 }\) ,centre the origin,which touches the curve \(C _ { 1 }\) at the points where \(x = \pm y\) .
6.(a)Given that $x ^ { 4 } + y ^ { 4 } = 1$ ,prove that $x ^ { 2 } + y ^ { 2 }$ is a maximum when $x = \pm y$ ,and find the maximum and minimum values of $x ^ { 2 } + y ^ { 2 }$ .\\
(b)On the same diagram,sketch the curves $C _ { 1 }$ and $C _ { 2 }$ with equations $x ^ { 4 } + y ^ { 4 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 1$ respectively.\\
(c)Write down the equation of the circle $C _ { 3 }$ ,centre the origin,which touches the curve $C _ { 1 }$ at the points where $x = \pm y$ .\\
\hfill \mbox{\textit{Edexcel AEA 2010 Q6 [10]}}