# Optimization calculus Problem

I have a calculus final coming up and was going over a practice exam my professor gave me and I came across a problem I was struggling with. I would post a picture but I am having trouble posting a picture So the problem reads.

The rectangle shown here has one side on the positive $x$-axis, one side on the positive $y$-axis, and its upper right hand vertex on the curve $y=e^{-x^2}$. What dimensions give the rectangle its largest area?

Keep In mind I already have the answer that’s not what I’m looking for, I’m searching for the process to arrive at the answer and the necessary work. BTW The answer is $x=1/\sqrt{2}$ and $y=e^{-1/2}$. Thank you for the help.

• 1) Draw a picture 2) Set up some parameters ($x, y$ here of course) 3) Identify the constraint ($y=e^{-x^2}$) here 4) Identify the quantity to optimize ($xy$ here) 5) Make the latter a function of one variable thanks to the constraint 6) Study that function to find the extremum – Julien Dec 13 '13 at 4:29
• Thank you for your reply it was very helpful, however I have one more question would finding the constraint on an equation like this entail finding the derivative of the curve's equation set to equal 0. I couldn’t figure out if that would give me the vertex of the curve,I.E. the upper right hand corner of the box. – user3097903 Dec 13 '13 at 4:42