Optimization - Rectangle Inscribed in a Parabola

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A rectangle is inscribed between the `x`-axis and a downward-opening parabola, as shown above. The parabola is described by the equation `y = -ax^2 + b` where both `a` and `b` are positive. You can reshape the rectangle by dragging the blue point at its lower-right corner.

Note: `x` is the distance from the origin to the lower-right corner of the rectangle; `x` is not the length of the base of the rectangle!

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  1. Let `a = 1` and `b = 7`. What value of `x` maximizes the area of the rectangle?
  2. Let `a = 1` and `b = 7`. What value of `x` maximizes the perimeter of the rectangle?
  3. Repeat the above two problems for `a` and `b` in general.
  4. From the previous exercise you can see that the `x` value where the perimeter is maximized depends only on the parameter `a`. Describe all parabolas that have an inscribed rectangle of maximum perimeter at `x = 1`.
  5. Occasionally it happens that for a given parabola the same value of `x` maximizes the area and the perimeter of the rectangle. If a parabola has this property, what is the relationship between `a` and `b`? Verify you findings by trying a few examples with the applet.