= Solution
The <flat function> $f$ is an <even function>, has its unique zero and <global minimum> at the origin, and approaches the horizontal <asymptote> $1$ as $x\to\pm\infty$. It decreases for negative $x$ and increases for positive $x$, meeting the origin with a horizontal <tangent line>.
Its <derivative> is an <odd function>: $f'(x)=2x^{-3}e^{-1/x^2}$ off zero, with $f'(0)=0$. It is negative to the left and positive to the right, tends to zero both at the origin and at infinity, and has one negative trough and one positive crest. Indeed $f''(x)=2e^{-1/x^2}(2-3x^2)/x^6$ away from zero, so the change in slope of $f'$ occurs once on each side. The exact crest and trough coordinates are not needed for the sketch.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1-flat-function.png]
{title=An even smooth flat function and its odd derivative, both meeting zero with all derivatives zero}
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