An example on real quantifier elimination

This instructive example has been used by Hoon Hong in a colloquium talk at the University of Passau in 2005.
Let , for given real numbers and . The graph shows parts of the (x,y)-plane in red where and in blue where . If both inequalities are true, the intersection of the red and blue domains is magenta. Question: For which and values will be the following statement true?

For all there exists such that and .

The geometrical meaning of this statement is clearly that the magenta domain always has a point for each value, that is, "horizontally infinite".

What positions of the sliders and ensure that the statement is true?

Select all that apply
  • A
  • B
  • C
  • D
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