Monotonic Function?

Question

Is ${tan}^{- 1} x$ $tan^-1x$ a monotonically increasing function for all $x$ $x$ belongs to $R$ $R$ ?

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Answer 1 · 2017-05-06T15:53:12

Yes. See the Explanation.

Let $f(x)=tan^-1x, x in RR.$

$:. f'(x)=1/(1+x^2).$

Now, $AA x in RR, x^2 ge 0 rArr 1+x^2 ge 1.$

$rArr f'(x)=1/(1+x^2) gt 0, aa x in RR.$

We coclude that, $f(x)=tan^-1x, x in RR" is "uarr.$

Answer 2 · 2017-05-06T15:54:32

Yes.

Yes it is, because deriving it

$d/(dx) arctan(x) = 1/(1+x^2) > 0$ so

$f(x) = int_a^x (d xi)/(1+ xi^2)$ is a monotonically increasing function from $a$ to $oo$