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?

Question

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&#039;t actually interpret the given data in a way that would produce the required conclusion, any ideas?

Oliver Shepherd · Accepted Answer

You get       x    2    −  2      |    x      |    ≤  f  (  x  )  ≤      x    2    +  2      |    x      |   not by squaring but by noting:      |    a      |    &amp;lt;      |    b      |   (or indeed   ≤) is the same as −b&amp;lt;a&amp;lt;bYou can then see from the behavior at 0 there can be no such functionCounter exampleAs mentioned in the comments   f  (  x  )  =      x    2  Missing bounds Even if it&#039;s for some range, like   [  0  ,  ∞  ) there can still be a function that isn&#039;t monotonic between the bounds. Eg x(x−1) (look at the roots, that&#039;s how I spotted this)

Proving that a function is strictly monotonic knowing that <mrow class="MJX-TeXAtom-ORD"> |

Answered question

Answer & Explanation

New Questions in Pre-Algebra