I have a question relating to calculus and inverse trigonometric functions. Any help is appreciated.

The question is:

The graphs of $y=ax$ and $y=\arctan(bx)$ intersect at three distinct points if?

A: $0<b<a$

B: $a<b<0$

C: $a=b$

D: $b<a<0$

The correct answer is D. Upon inserting the graphs into desmos I can visually see how they intersect at three distinct points but I am struggling to understand why.

Thank you for sharing your knowledge!

  • 1
    $\begingroup$ These solutions are not going to be elementary, so you're not going to be able to reasonably construct them in terms of $a$ and $b$. Do you know other tools for counting roots, like intermediate value theorem and/or mean value theorem (or Rolle's theorem)? $\endgroup$ May 26 at 11:55
  • $\begingroup$ The graphs will intersect in three distinct points as long as b< a. $\endgroup$ May 26 at 11:59
  • $\begingroup$ @GeorgeIvey Thank you! Not sure if the explanation will be too complicated but why is that so? $\endgroup$
    – irene
    May 26 at 12:01
  • $\begingroup$ @TheoBendit Hmm unfortunately not, closest thing I can think of is to do with polynomial roots, thanks a lot anyways! $\endgroup$
    – irene
    May 26 at 12:02
  • 1
    $\begingroup$ First thing is that both a and b should be of the same sign for the 3 roots condition to hold ( else there will be only 1 root )..secondly the relation between a and b can be obtained by comparing the slope of both the curves at origin $\endgroup$
    – user1012971
    May 26 at 12:58

1 Answer 1


Note that: $$\frac{d}{dx}(tan^{-1}(bx))|_{x=0} = \bigg[\frac{1}{1+(bx)^2}\cdot b\bigg]\bigg|_{x=0} = b$$ and $$\frac{d}{dx}(ax)|_{x=0} = a.$$ We want the slope of the tangent line at the origin to be steeper for the arctan function so that the other line intersects it in 3 points instead of 1 (recall the graph of artan is just an infinitely extended, bounded "S" shape).

  • $\begingroup$ Also,we can comment that both a and b are of the same sign for the curves to intersect at three different points $\endgroup$
    – user1012971
    May 26 at 13:01
  • 1
    $\begingroup$ Don't forget it's a multiple choice question. You only have to select the right answer - not prove anything. Everyone knows that the tangent of a very small angle is equal to the angle itself. (Don't they?) That means the slope of arctan(bx) at the origin is b. Select answer D and on to the next question. $\endgroup$
    – stretch
    May 26 at 13:29

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