Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 1 / 11 / Solution

Differentiation and factorization give $f^{\prime }(x)=e^{-x^2}(1+x^2-2x^4)=e^{-x^2}(1-x^2)(1+2x^2)$. The exponential and $1+2x^2$ are positive, so the only stationary points are at $x=\pm 1$. The derivative is positive for $|x|<1$ and negative for $|x|>1$, givingThe function is odd, crosses the origin with slope one, and tends to zero from below as $x\to -\infty$ and from above as $x\to \infty$. These facts determine the sketch.