# Advent calendar 2019

## 10 December

For all values of \(x\), the function \(f(x)=ax+b\) satisfies

$$8x-8-x^2\leqslant f(x)\leqslant x^2.$$

What is \(f(65)\)?

Edit: The left-hand quadratic originally said \(8-8x-x^2\). This was a typo and has now been corrected.

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If we plot the two quadratics, we see that they meet at \(x=2\) where they both have gradient 4.

The line \(f(x)\) must therefore pass through the point \((2,4)\) and have gradient 4, so its equation is \(f(x)=4x-4\).

This means that \(f(65)\) is **256**.