Review question

# If the gradient is $x(1-x^2)e^{-x^2}$ can we find the stationary points? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6620

## Suggestion

Show that $C_1$ has a minimum point at the origin and a maximum point at $(1,\frac{1}{2}e^{-1})$. Find the coordinates of the other stationary point. Give a rough sketch of $C_1$.

What condition is required for a stationary point, and how do we classify them?

To find the coordinates of the maxima, we need to know $y$ as a function of $x$. What form do you think the function will take?

Other things to consider when sketching this function:

1. what happens as $x \rightarrow \pm\infty$?
2. is this function even or odd (or neither)?

Show that $C_2$ has a minimum point at the origin and a maximum point at $(1,k)$, where $k>\frac{1}{2}e^{-1}$.

Can we find the $x$ coordinates of the stationary points? Can we classify them?

Can we prove the inequality involving $k$ by comparing with the function $C_1$?