Show that this equation only has one solution

[Kqwert]

Homework Statement

Show that the equation arctan(x) = x^2 have at least one solution. Then argue why the equation only has one, positive solution r.

Homework Equations

The Attempt at a Solution

I know how to prove that the equation has at least one solution (IVT), but I do not know how to show the second part. If the derivative is strictly positive/negative I know the answer, but that is not the case here.

 

[kuruman]

What is the one solution that you found? What happens at values less than that? Plotting arctan(x) and x² on the same graph would be useful to guide your thinking.

 

[Kqwert]

I re-wrote the function to h(x) = arctan(x) - x^2 and found that the function h(x) had a zero in the interval [-2,0.5].

 

[hilbert2]

Suggestions:

Argue that $x^2$ and $\arctan x$ are of different sign for $x<0$, so there is no negative solution.

For $x>0$ both $x^2$ and $\arctan x$ are increasing functions, one of them has positive second derivative for all positive values of $x$, the other one a negative second derivative for all positive $x$. Why does this lead to the graphs of $x^2$ and $\arctan x$ crossing at only one point?

 

[Kqwert]

So one of the functions has an increasing derivative while the other has a decreasing derivative. Not sure why that results in them only crossing once though..?

 

[hilbert2]

If you have two objects initially at the same position on x-axis, and one of them accelerating after that and the other one decelerating, it it possible that there is more than one instant when they're again at same position on x-axis?

 

[Kqwert]

No, I agree with that. But why not use the first derivative, which is common in these tasks? (Tasks were you are supposed to show that an eq. only has one solution)

 

[epenguin]

Have you done as advised here?

Plotting arctan(x) and x² on the same graph would be useful to guide your thinking.

Is your problem that you haven't done this, or is it just that you don't know how to turn what is then obvious into the required formal statements?
Maybe if you do preferably sketch, otherwise plot it, it will come back to you that actually arctan(x) against x is not one single curve but an infinity of them, however this doesn't really change anything except for making your argument complete and correct.

 

[Kqwert]

Have you done as advised here? Is your problem that you haven't done this, or is it just that you don't know how to turn what is then obvious into the required formal statements?
Maybe if you do preferably sketch, otherwise plot it, it will come back to you that actually arctan(x) against x is not one single curve but an infinity of them, however this doesn't really change anything except for making your argument complete and correct.

I have plotted them, and notice that there are only two intercepts - 0 and another value slightly below 1, but not sure how I should turn this into a formal statement?

 

[epenguin]

I have plotted them, and notice that there are only two intercepts - 0 and another value slightly below 1, but not sure how I should turn this into a formal statement?

What is the slope of each function at x = 0 ?

Then between #4 and #5 I think you have it.