Puzzle 5: The sandwiched quadratic
To solve this puzzle, you need to find a clue in (virtual) Edinburgh: On the Royal Mile, there is a bus stop. \(E\) is the number of the bus going towards Dumfries.
You know that \(f\) is a quadratic, and so can be written as \(f(x)=ax^2+bx+c\) for some real numbers \(a\), \(b\), and \(c\); but you've forgetten exactly which quadratic it is. You remember that for all real values of \(x\), \(f\) satisfies
$$\tfrac{1}{4}x^2+2x-8\leqslant f(x)\leqslant(x-2)^2.$$You also remember that the minimum value of \(f\) is at \(x=0\).
What is \(f(E)\)?
The answer to this puzzle is a four-digit number that scores 4 points.
Read the instructions for a reminder of what these points mean.
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