Question

how do you solve arctan(x) + x/(x^2+1)?

Answer 1

I know the answer is x=0 I just don't know how to get there

Answer 2

There is nothing to solve.

Answer 3

arctan(x) + x/(x^2+1)=0 is what I meant actually sorry

Answer 4

I can change it so it looks like x=tan(-x/(x^2+1)), but I'm not sure that helps

Answer 5

Is this one of those problems where you have to integrate by parts?

Answer 6

this problem is to find the critical points for x*arctan(x)

Answer 7

Let \( f(x) = \arctan(x) + x/(x^2 + 1) \).

Then
\[ f'(x) = \frac{1}{x^2+1} + \frac{1-x^2}{(x^2+1)^2} = \frac{2}{(x^2+1)^2} > 0 \ \ \hbox{for all} \ \ x \]
Therefore the function is strictly monotonically increasing.

Now f(0) = 0.

Given these facts can there be any other \( x \neq 0 \) such that \( f(x) = 0\) ?

Answer 8

is there any way to use trig to simplify this one?

Answer 9

I've just given you answer:
- f(x) is an everywhere strictly increasing function
- f(0) = 0

Given these two facts about f, is it possible there can be another solution? i.e., another x not equal to zero such f(x) = 0?

Answer 10

What class are you taking? Have you had calculus?

Answer 11

how can you know that x=0?
I am in calculus right now

Answer 12

other than subbing in 0 and checking

Answer 13

arctan(0) = 0 and if x = 0, then certainly x/(x^2 + 1) = 0. Therefore f(0) = 0.

Answer 14

how did you know to check at x=0?

Answer 15

Because it was obvious value to check. Good luck to you.

Answer 16

ok, thanks

Answer 17

So could you post the original problem just as it appears in your textbook>

Answer 18

I'm looking for the critical points of x*arctan(x)

Answer 19

so when the derivative is 0
arctan(x) + x/(x^2+1)=0

I know the answer is 0 because I can sub it in and it works, I just wanted to know the way to simplify the probem showing that the derivative is 0 when x=0

Answer 20

Ok. So for g(x) = x.arctan(x), g'(x) = 0 if and only if x = 0. At that critical value, g(0) = 0.

Further, g(x) is an even function, g(x) = g(-x); and for x > 0, g(x) > 0. Therefore g(x) has a minimum at x = 0.

Answer 21

Ok, I think that is enough justification for x=0,
all I wanted to know was if there was a way to simplify arctan(x) + x/(x^2+1)=0 to solve for x

Answer 22

The answer is there is no way to solve this problem analytically, other than the one I've given. x=0 works, but if you didn't think to guess that then you might be stuck for a while trying to figure it out.

Answer 23

alright, thanks for your help