 Solution

ID: 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

f^{'} (x) = e^{- x^{2}} (1 + x^{2} - 2 x^{4}) = e^{- x^{2}} (1 - x^{2}) (1 + 2 x^{2})

. The exponential and

1 + 2 x^{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

(- 1, - 2/ e) a global minimum, (1, 2/ e) a global maximum .

The 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.

Stationary points and Gaussian decay of the odd polynomial-exponential profile

. Original plot of

(x + x^{3}) e^{- x^{2}}

, with its global extrema marked. Shading indicates the equal and opposite signed areas on the two half-lines. Details.

 New to topics? Read the docs here!