2024 THREE(e)

The diagram below shows part of the graph of the function $f(x)=e^{-x^2}$, where $x\ge 0$.

The point P lies on the curve and the point Q lies on the $x$-axis so that OP = PQ, where O is
the origin.

Prove that the largest possible area of the triangle OPQ is $\frac{1}{\sqrt{2e}}$.

*You do not need to show that the area you have found is a maximum.*

*You must use calculus and show any derivatives that you need to find when solving this problem.*