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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.

Diagram

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.

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How did you make that picture? –  yiyi Oct 26 '12 at 1:06
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1 Answer

up vote 8 down vote accepted

You found the height of the triangle correctly. It simplifies to $(a+1)e^{-a}$.

To find the base of the triangle, put $y=0$ in the equation of the tangent line, and solve for $x$. There is very nice cancellation, and we get $x=a+1$. Thus the area $A(a)$, as a function of $a$, is given by $$A(a)=\frac{1}{2}(a+1)^2 e^{-a}.$$ There should now be no difficulty in maximizing the area. (I would prefer to maximize $2A(a)$, no fractions.) Calculate the derivative with respect to the variable $a$ as usual, and argue that one of the two critical points gives maximum area. We have to be a little careful about the geometry: if $a< -1$, there is no triangle in the first quadrant. The formula for $A(a)$ cheerfully gives you an "area" even when $a<-1$.

Comment: Nice picture! You have clearly tried to understand the problem. All too many people try to solve problems using mechanical manipulation only. That usually works for simple enough problems, but breaks down for anything more subtle.

There is a typo in the picture. The equation of the tangent line at the point $(a,e^{-a})$ should be $y=e^{-a}+f'(a)(x-a)$.

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Thanks a bunch! –  NehriMattise Nov 17 '11 at 7:27
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