Proof that |arctan⁡(x)−π4−(x−1)2|≤(x−1)22I'm trying to prove that, for every x≥1 :|arctan⁡(x)−π4−(x−1)2|≤(x−1)22I could do it graphically on R, but how to make a formal algebraic proof?

Name: Proof that |arctan⁡(x)−π4−(x−1)2|≤(x−1)22I&#039;m trying to prove that, for every x≥1 :|arctan⁡(x)−π4−(x−1)2|≤(x−1)22I could do it graphically on R, but how to make a formal algebraic proof?
Uploaded: 2022-02-23T17:22:57+00:00
Description: Proof that |arctan⁡(x)−π4−(x−1)2|≤(x−1)22I&#039;m trying to prove that, for every x≥1 :|arctan⁡(x)−π4−(x−1)2|≤(x−1)22I could do it graphically on R, but how to make a formal algebraic proof?

Question

Proof that |arctan⁡(x)−π4−(x−1)2|≤(x−1)22I&#039;m trying to prove that, for every x≥1 :|arctan⁡(x)−π4−(x−1)2|≤(x−1)22I could do it graphically on R, but how to make a formal algebraic proof?

chaloideq1 · Accepted Answer

We have that for x≥1arctan⁡x−π4−(x−1)2≤0⇒|arctan⁡x−π4−(x−1)2|=π4+(x−1)2−arctan⁡xthen considerf(x)=(x−1)22+arctan⁡x−π4−(x−1)2⇒f′(x)=1x2+1+x−32≥0and since f(1)=0 we have that f(x)≥0 and the inequality is proved.

euromillionsna · Accepted Answer

Hint: For each x&amp;gt;1, Taylor gives  arctan⁡(x)=π4+(x−1)2+arctan⁡(cx)2(x−1)2  where 1&amp;lt;cx&amp;lt;x Thus all you need to show is that |arctan⁡(c)|≤1 for all c≥1