How can I demonstrate that if \[ g\colon\mathbb R\to \mathbb R, x \mapsto f(x) + \frac 12\arctan{\sqrt{x+1}} \] is constant, where \[ f\colon\mathbb R\to\mathbb R, x\mapsto \arctan(x+2) - \arctan x. \]
demonstrate the continuity of a function
1
$\begingroup$
analysis
-
0You have the wrong domain for $g$. – 2012-09-20
1 Answers
1
It is not constant.
Differentiate to get $g'(x) = \frac{1}{(x+2)^2+1} - \frac{1}{x^2+1}+\frac{1}{4 \sqrt{x+1}(x+2)}$. $g'(0) = -\frac{27}{4}$, hence it is not constant.