Proving that a function is strictly monotonic knowing that | f ( x ) −...

gaiaecologicaq2

gaiaecologicaq2

Answered

2022-07-09

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

Answer & Explanation

Oliver Shepherd

Oliver Shepherd

Expert

2022-07-10Added 24 answers

You get x 2 2 | x | f ( x ) x 2 + 2 | x | not by squaring but by noting:
| a | < | b | (or indeed ) 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 ( x ) = x 2
Missing bounds Even if it's for some range, like [ 0 , ) 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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