Finding existence of zeros of cubic by multiplying y values?

In summary, the question asked for a function that has only one root when the coefficient of x^3 is greater than zero. Andrew found two stationary points where the y value was greater than zero. The first stationary point was at +1/sqrt(3) and the second stationary point was at -1/sqrt(3). The y values at these two stationary points were then multiplied together to find the root of the function. If the x-axis was located at the local maximum or local minimum point of the function's graph, then there would be only one root.
  • #1
Celestion
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Homework Statement



I can do the question, but in a different way to the worked solution which I don't understand. So my question is can anyone explain the worked solution which is in point 3 below.

The question was to show there is exactly one zero to the function f(x) = Ax^3 - Ax + 1, with A>0, when A < 3sqrt(3) / 2.

Homework Equations



The part above this asks to find the stationary points, and there are two of them, at + and - 1/sqrt(3).

The Attempt at a Solution



My reasoning was that since A>0, which is the coefficient of x^3 (i.e. the highest degree term), the graph goes up into the top right of the first quadrant as x and y approach positive infinity. And therefore to have only one root, the y value of the stationary point at x = 1/sqrt(3) must be >0. (I also considered the case where the stationary point at 1/sqrt(3) was <0 and the one at -1/sqrt(3) was also <0, but this is impossible given the constant term of +1 and from playing with graphing software, and having A>0). Solving this, i.e. y = f(x) being >0 at x=1/sqrt(3) gives the correct answer of A < 3sqrt(3) / 2 from point 1 above.

HOWEVER, the worked solution calculates the y values at both + and - 1/sqrt(3), and then multiplies the y values together, giving a quadratic in A (i.e. containing A^2), and then uses the condition that the two y values multiplied together must be >0 for there to be only one zero. Which I don't understand at all...

i.e. f(1/sqrt(3)) * f(-1/sqrt(3)) = (1 - 2A/3sqrt(3))(1 + 2A/3sqrt(3)) = 1 - 4A^2 / 27

and when this is >0, i.e. when A < 3sqrt(3)/2, there will be only one zero of f(x).
 
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  • #2
Draw a graph, without scale or axes, of a cubic of this sort. It'll come from ##-\infty## at the left, rising but curving downwards, then reach a maximum at ##x=x_1=-\frac1{\sqrt 3}, y=y_1##, curve down, go through an inflexion point at ##x=0##, start curving up, reach a minimum at ##x=x_2=\frac1{\sqrt 3},y=y_2##, then keep curving up towards ##+\infty##.

Now imagine that you can draw the x-axis anywhere on the page. The number of roots is the number of times the axis cuts the curve.

If the x-axis is tangent to the curve at the local maximum point ##(x_1,y_1)##, there are two roots. If it is higher than that, there is only one. If it is tangent to the curve at the local min point ##(x_2,y_2)## there are two roots. If it is lower than that, there is only one. If it is between the heights of the local minimum and the local maximum it cuts the curve three times.

So there is one root iff the x-axis cuts the curve below ##y_2## or above ##y_1##, which is the case iff the signs of ##y_1## and ##y_2## are the same, which is the case iff ##y_1\times y_2>0##.

I think your solution is just as good though.
 
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  • #3
Thanks Andrew, that's great, it makes perfect sense. I think their/your solution is more elegant than mine :) now that I understand it
 

1. How do you find the existence of zeros of a cubic by multiplying y values?

The existence of zeros of a cubic can be found by graphing the function and looking for the x-intercepts, which represent the zeros. To find the x-intercepts, you can multiply the y-values on either side of the x-axis until you get a value close to zero. This will help you determine the approximate location of the zeros.

2. Can you use any method other than multiplying y values to find the zeros of a cubic?

Yes, there are several methods to find the zeros of a cubic. Some common methods include factoring, using the quadratic formula, and using the rational roots theorem. However, multiplying y values can also be an effective method, especially if you are graphing the function.

3. Is it possible for a cubic function to have more than three zeros?

No, a cubic function can only have three zeros. This is because a cubic function is a polynomial of degree 3, and the number of zeros of a polynomial is always equal to its degree.

4. How can you determine the multiplicity of a zero when using the method of multiplying y values?

The multiplicity of a zero can be determined by counting the number of times the y-values are multiplied to get close to zero. For example, if you need to multiply the y-values five times to get close to zero, then the multiplicity of that zero is 5.

5. What is the significance of finding the zeros of a cubic function?

The zeros of a cubic function represent the x-values where the function crosses the x-axis. This helps us identify the roots or solutions of the function, which can be useful for solving equations or understanding the behavior of the function. Additionally, the zeros can also provide information about the shape and symmetry of the graph.

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