# Show that this equation only has one solution

• Kqwert
In summary, the conversation discusses how to prove that the equation arctan(x) = x^2 has at least one solution through the use of the Intermediate Value Theorem. It also addresses the issue of why the equation only has one, positive solution r. It is argued that for x<0, the functions arctan(x) and x^2 are of different sign, and for x>0, both functions are increasing with one having a positive second derivative and the other a negative second derivative. This results in the graphs of the two functions only crossing at one point. The conversation also suggests plotting the two functions to better understand this concept.
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.

## 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.

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.

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].

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?

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..?

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?

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)

Have you done as advised here?
kuruman said:
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.

epenguin said:
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?

Kqwert said:
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.

## 1. What does it mean for an equation to have only one solution?

Having only one solution means that there is only one value for the variable in the equation that satisfies the equation and makes it true.

## 2. How can I tell if an equation only has one solution?

An equation can only have one solution if it is a linear equation with one variable, or if it is a quadratic equation with a discriminant of zero.

## 3. Why is it important to show that an equation only has one solution?

Knowing that an equation only has one solution can help us determine the uniqueness of a problem or situation. It can also help us make accurate predictions and solve problems more efficiently.

## 4. What are some methods for showing that an equation has only one solution?

One method is to graph the equation and see if it only intersects the x-axis at one point. Another method is to use algebraic manipulations to isolate the variable and see if there is only one value that satisfies the equation.

## 5. Can an equation have only one solution in some cases but multiple solutions in others?

Yes, this is possible. For example, a quadratic equation can have one solution if its discriminant is zero, but it can also have two solutions if the discriminant is positive. It depends on the specific equation and its coefficients.

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