Maximum Area

[janeann]

Suppose a rectangle is in base of the x-axis and is inscribed in the curve f(x)=(1)/(1+x^2). Find the largest such rectangle. Is there anything geometrically intresting about the critical points.

I'm not sure where to even begin for this problem.Should I try and scetch the curve first?

 

[galactus]

This function is symmetric about the y-axis, so center your rectangle at the origin.

The length of the base is 2x and the height is y.

\(\displaystyle A=2xy\)

But, \(\displaystyle y=\frac{1}{1+x^{2}}\)

Sub this into A. Now, we are in terms of one variable....x

\(\displaystyle A=\frac{2x}{1+x^{2}}\)

Now, differentiate, set to 0 and solve for x. y follows

The area follows from x and y.

 

[janeann]

as my derivative of i got (-2x)/(1+x^2)^2 is that correct, and if it is how would i solve it when i set it equal to 0?

 

[JeffM]

as my derivative of i got (-2x)/(1+x^2)^2 is that correct, No, it is not correct. What is the rule for differentiating the quotient (u / v) where both u and v are functions of x?

and if it is how would i solve it when i set it equal to 0? Let's worry about that when we have the correct derivative