= Solution
This <exponential function> is positive everywhere, with $y(0)=1$, $y\to\infty$ as $x\to-\infty$ and $y\to0$ as $x\to\infty$. Its <derivatives> are
$$
y'=-3x^2e^{-x^3}\leq0,\qquad
y''=3x(3x^3-2)e^{-x^3}.
$$
It is strictly decreasing overall, with a horizontal stationary <inflection point> at \b[$(0,1)$]. A second, nonstationary <inflection point> occurs at $x=(2/3)^{1/3}$, where $y=e^{-2/3}$. The only horizontal <asymptote> is \b[$y=0$ toward positive infinity]; the graph is not symmetric about the vertical axis.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1-cubic-exponential.png]
{title=The decreasing cubic exponential, its stationary inflection, and its positive horizontal asymptote}
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