I am currently struggling with a math problem and was wondering if someone could help me out. The problem is as follows:
In a coordinate system, a rectangle is drawn with one corner at the origin, one corner on the positive $x$-axis, one corner on the positive $y$-axis, and one corner on the graph of the function $f(x)=\mathrm{e}^{-5 x}$.
I need to determine the largest possible area that such a rectangle can have. I have tried to solve it using basic calculus, but I am not getting the correct answer. Could someone provide me with some guidance or a step-by-step solution to this problem?
Thank you in advance for your help!