gaiaecologicaq2

2022-07-09

Proving that a function is strictly monotonic knowing that $|f\left(x\right)-{x}^{2}|\le 2|x|$
Prove that the following real-valued function is strictly monotonic, knowing that $|f\left(x\right)-{x}^{2}|\le 2|x|$.
I can't actually interpret the given data in a way that would produce the required conclusion, any ideas?

Oliver Shepherd

Expert

You get ${x}^{2}-2|x|\le f\left(x\right)\le {x}^{2}+2|x|$ not by squaring but by noting:
$|a|<|b|$ (or indeed $\le$) is the same as −b<a<bYou can then see from the behavior at 0 there can be no such function
Counter example
As mentioned in the comments $f\left(x\right)={x}^{2}$
Missing bounds Even if it's for some range, like $\left[0,\mathrm{\infty }\right)$ there can still be a function that isn't monotonic between the bounds. Eg x(x−1) (look at the roots, that's how I spotted this)

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