Answer:
Step-by-step explanation:
<u>Optimizing With Derivatives
</u>
One of the most-used applications of derivatives is to maximize or minimize functions. We need to recall that if f(x) is a real function and f'(x) is the derivative of f, then we can find the critical points of f by setting
Then we must test the critical points in the second derivative f''(x) and if
f''(x) is positive, then x is a minimum
f''(x) is negative, then x is a maximum
The problem requires us to find the maximum area of the rectangle which base is x and height is f(x), where
The area of the rectangle is the product of the base by the height, so
Let's find the first derivative
Setting A'=0
Solving for x
Let's compute the second derivative
Factoring
Evaluating for the critical point we can see the first factor (2x) is positive. The exponential is always positive, we only need to find the sign of
Since the expression is negative, thus
A''(x)<0 and the critical point is a maximum
The maximum area is