最優化問題 (１)

Let the base of the semi-circle be the x-axis.

The perpendicular line of the x-axis, which intersect with the x-axis at the mid-point is y-axis.

The formula restricts x and y is: x^2+y^2=4; y>0
We need to find the maximum value of |x|y.

1. Express y in terms of x: y=sqrt(4-x^2)

2. Substitute the value of y: xy=x[sqrt(4-x^2)]

3. Differentiate it: d/dx x[sqrt(4-x^2)] = [sqrt(4-x^2)]*[1-(x^2)/(4-x^2)]

4. Make the result be zero. [sqrt(4-x^2)]*[1-(x^2)/(4-x^2)]=0

Solve it: x=sqrt(2) or -sqrt(2)

So the solution is the length of the rectangle is equal to sqrt(2)-[-sqrt(2)]=2sqrt(2)

The height is sqrt[4-(sqrt(2))^2]=sqrt(2)

The area is 2(sqrt(2))*sqrt(2)=4.