A right triangle has one leg on the x-axis.The vertex at the right end of that leg is at the point (3,0).The other vertex touches the graph of y=e^x.The entire triangle is to lie in the first quadrant.Find the max area of this triangle.

Thanks.

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- Mar 6th 2010, 05:57 PM #1

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## Help with max and min.

A right triangle has one leg on the x-axis.The vertex at the right end of that leg is at the point (3,0).The other vertex touches the graph of y=e^x.The entire triangle is to lie in the first quadrant.Find the max area of this triangle.

Thanks.

- Mar 6th 2010, 08:19 PM #2
From what you have described, it would appear that your triangle has vertices:

and I assume that .

Therefore its base is and its height is .

So its area is given by the formula:

.

Since you want the maximum area:

.

Setting the derivative equal to and solving for :

or .

But since for all , that means that only is valid.

Therefore .

To check that this IS a maximum, find the second derivative and check that it is negative at .

At

.

This function is negative at , so does indeed maximise the area.

So at :

square units.

- Mar 13th 2010, 12:17 PM #3