(i) The curve C1 passes through the origin in the x-y plane and
its gradient is given by dy/dx = x(1 - x^2)e^(-x^2). Show that C1
has a minimum point at the origin and a maximum point at (1, 1/2e^(-1)). Find the coordinates of the other stationary point. Give a
rough sketch of C1. (ii) The curve C2 passes through the origin
and its gradient is given by dy/dx = x(1 - x^2)e^(-x^3). Show that
C2 has a minimum point at the origin and a maximum point at (1, k),
where k > 1/2e^(-1). (You need not find k.) Comments: No work is
required to find the x-coordinate of the stationary points, but you
have to integrate the differential equation to find the y-coordinate. For the second part, you cannot integrate the equation
— other than numerically, or in terms of rather obscure special
functions that you almost certainly haven't come across, such as
the incomplete gamma function defined by Γ(x, a) = ∫(a, 0) t^(x-1)e^(-t)dt. However, you can obtain an estimate, which is all that is
required, by comparing the gradients of C1 with C2 and thinking of
the graphs for -1 ≤ x ≤ 1. This is perhaps a bit tricky; an idea
that you may well not alight on under examination conditions.