Challenging +1.8 This question requires implicit differentiation to find dy/dx, then a second implicit differentiation to find d²y/dx², involving careful application of the chain rule and product rule. The first part requires showing dy/dx is always finite (denominator never zero), which needs algebraic manipulation. The second part involves substantial algebraic computation with the specific point, but follows a standard procedure. The complexity lies in the multi-step calculation and algebraic manipulation rather than novel conceptual insight.
10 The curve \(C\) has equation
$$y = x ^ { 2 } + \lambda \sin ( x + y ) ,$$
where \(\lambda\) is a constant, and passes through the point \(A \left( \frac { 1 } { 4 } \pi , \frac { 1 } { 4 } \pi \right)\). Show that \(C\) has no tangent which is parallel to the \(y\)-axis.
Show that, at \(A\),
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 - \frac { 1 } { 64 } \pi ( 4 - \pi ) ( \pi + 2 ) ^ { 2 }$$
10 The curve $C$ has equation
$$y = x ^ { 2 } + \lambda \sin ( x + y ) ,$$
where $\lambda$ is a constant, and passes through the point $A \left( \frac { 1 } { 4 } \pi , \frac { 1 } { 4 } \pi \right)$. Show that $C$ has no tangent which is parallel to the $y$-axis.
Show that, at $A$,
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 - \frac { 1 } { 64 } \pi ( 4 - \pi ) ( \pi + 2 ) ^ { 2 }$$
\hfill \mbox{\textit{CAIE FP1 2006 Q10 [10]}}