I'm stuck on a question involving maximizing the area of a triangle:
What is the area of the largest triangle that can be formed in the first quadrant by the x-axis, the y-axis, and a tangent line to the graph $y=e^{-x}$ ?
I made a diagram for myself in order to better understand the question. Following is what I have come up with so far.
So I am looking to maximize the area of the triangle that is shaded in light blue. The area of the triangle is of course $A=0.5 bh$. The height will be the value of the tangent line where $x=0$, and the base of the triangle will be the length from the y-axis to the point where the tangent line intercepts the x-axis. I have found the tangent line for the point $a$:
$y=e^{-a}-e^{-a}(x-a)$
So the height of the triangle should be $y=e^{-a}-e^{-a}(0-a)$. I am unsure how to proceed and find the length from the y-axis to the point where the tangent line intercepts the x-axis (the inverse of the tangent line?). I believe that after I find the equation for the area of the triangle I will need to the find the maximum value.
Thank you.