2001 Paper 2 Q5

Year: 2001
Paper: 2
Question Number: 5

Course: LFM Pure
Section: Differential equations

Difficulty: 1600.0 Banger: 1485.5

differential equation curve sketching stationary points integration inequality first order ODE exponential function maxima and minima

Problem

The curve $C_1$ passes through the origin in the $x$--$y$ plane and its gradient is given by $$ \frac{\d y}{\d x} =x(1-x^2)\e^{-x^2}. $$ Show that $C_1$ has a minimum point at the origin and a maximum point at $\left(1,{\frac12\, \e^{-1}} \right)$. Find the coordinates of the other stationary point. Give a rough sketch of $C_1$. The curve $C_2$ passes through the origin and its gradient is given by $$ \frac{\d y}{\d x}= x(1-x^2)\e^{-x^3}. $$ Show that $C_2$ has a minimum point at the origin and a maximum point at $(1,k)$, where \phantom{} $k > \frac12 \,\e^{-1}.$ (You need not find $k$.)

No solution available for this problem.

Rating Information

Difficulty Rating: 1600.0

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Banger Rating: 1485.5

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Problem source

The curve $C_1$ passes through the origin in the $x$--$y$ plane
and its gradient is given by
$$
\frac{\d y}{\d x} =x(1-x^2)\e^{-x^2}.
$$
Show that
$C_1$ has a minimum point at the origin and
a maximum point at 
$\left(1,{\frac12\, \e^{-1}} \right)$. Find the coordinates of the other stationary point.
Give a rough sketch of $C_1$.

The curve $C_2$ passes through the origin and its gradient is given by
$$
\frac{\d y}{\d x}=
x(1-x^2)\e^{-x^3}.
$$
Show that $C_2$ has a minimum point at the origin and a maximum point at $(1,k)$, where \phantom{}
$k >  \frac12 \,\e^{-1}.$ (You need not find $k$.)

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