Optimization hw problem

Question

 A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y =1-x^2. What are the dimensions of such a rectangle with the greatest possible area?

Width = 

Height = 

Answer 1

Let (x, y) is the vertex of rectangle in 1st quadrant. Then y = 1 - x² and area A(x) = 2xy = 2x(1 - x²) = 2x - 2x^3;

A'(x) = 2 - 6x² = 0; th2n x = 1/√3; 2x = 2/√3 and y = 1 - 1/3 = 2/3.

To prove that this is maximum area dimentions: A''(x) = - 12x; A''(1/√3) = - 12/√3 < 0.

Answer 2

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