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Largest possible area for a rectangle bordered by the x-axis

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I am in Freshmen Level Calculus. We are in section 4.5 of the Salas, Hille, and Etgen book, "Calculus: One and Several Variables" 9th edition.

The name of the chapter is The Mean value theorem and its applications, But from my understanding of the examples, It does not use the MVT. The previous sections were about local extreme values, endpoint and absolute extreme values. This section is entitled "Some Max-Min Problems".

The problem is as follows:
4. Find the largest possible area for a rectangle with base on the x-axis and upper verticies on the curve y = 4 minus x squared.

Okay, I tried to work it out the same way the example did it.
I wrote two equations, Area = 2xy (Two x is the base on the x axis, the curve and the rectangle placed on a coordinate plane.) and y = 4 - x squared.

I tried to put the area in the form of x only, substituting the curve equation for y. Then I differentiated and set f prime equal to zero.

I got a radical 3 something in my answer, and it doesn't look like I'm doing it right at all. I also dont understand how to get the boundaried for testing the endpoints... In the examples they always derived a closed interval for x.

I hope this was enough detail. I need to know If I am working this out correctly or if I am missing some part of the problem... thank you.

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The largest possible area for a rectangle bordered by the x-axis is determined.

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