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

## Question

The curve $C_1$ passes through the origin in the $x\text{-}y$ plane and its gradient is given by $\frac{dy}{dx}=x(1-x^2)e^{-x^2}.$ 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 $C_1$.

The curve $C_2$ passes through the origin in the $x\text{-}y$ plane and its gradient is given by $\frac{dy}{dx}=x(1-x^2)e^{-x^3}.$ Show that $C_2$ has a minimum point at the origin and a maximum point at $(1,k)$, where $k>\frac{1}{2}e^{-1}$. (You need not find $k$.)