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I was checking the answers of the coordinate axes has minimum area question.

In short, the mentioned question asks the minimum area formed by a tangent to $y=4-x^2$ and coordinate axes. I noticed one of the answers to the question mentioned an interesting property:

Using similar triangles, it can be proven that the area of the triangle is twice the product $xy$ where $y=f(x)$.

This property seems valid for the mentioned question, as well as $y=a/x$ and several other forms that I tried. However, I cannot prove or disprove.

Can we prove or disprove the property? In what conditions is the property valid?

Kartal Tabak
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    I don't think the statement is true. Take a tangent to $y=4-x^2$ at the point $(2,0).$ The triangle has vertices $(0,0),$ $(2,0),$ and $(0,8),$ with area $8$, but $xy=0.$ – David K Feb 07 '21 at 00:00
  • @David well, the triangle is degenerate at that point. the slope is zero, and the tangent crosses x-axis at infinity. However, your case seems to be true: $xy=0$, if the area calculation was correct, then area would be calculated as infinity. – Kartal Tabak Feb 07 '21 at 01:20
  • I asked a follow-up question: https://math.stackexchange.com/questions/4015695/the-method-to-calculate-area-between-fx-and-coordinate-axes – Kartal Tabak Feb 07 '21 at 01:20
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    You're thinking of the triangle with vertex at $(0,4).$ I considered using that one, since it's a more dramatic example, but I did not want to deal with degenerate triangles. At $(2,0)$ the slope is $-4$ so we get an ordinary (but still too large) triangle. – David K Feb 07 '21 at 01:43
  • Yes, you are right both against your counter-example, and how I misread your comment. – Kartal Tabak Feb 07 '21 at 01:51

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In another answer from your link it is proven that: $$ A=\frac{(4+x^2)^2}{4x}\ne2x(4-x^2). $$

user
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  • Ok, the area formulation is not aligned. However, the algorithm seems to find the optimal solution. I have posted a follow-up question: https://math.stackexchange.com/questions/4015695/the-method-to-calculate-area-between-fx-and-coordinate-axes – Kartal Tabak Feb 07 '21 at 01:16