How can cosx = x have more than one solution?

[Jan Hill]

Homework Statement

How can cosx = x have more than one solution?

Homework Equations

The Attempt at a Solution

I don't know how to begin this.

 

[Mark44]

Why do you think it might?

 

[Jan Hill]

because of the wave nature of the graph of cosx

 

[Mark44]

Sketch graphs of y = cos(x) and y = x. At how many points do these graphs intersect?

As an alternative you can graph y = cos(x) - x. The x-intercepts of this graph indicate the values for which cos(x) = x.

 

[Jan Hill]

the graphs intersect at more than 1 point, in fact at 3 points and does that mean we can say that that is why cosx = x has more than one solution because is more than one value for x at which cos x equals it?

 

[Mark44]

The only intersection point I see is for x ~ .72. What did you graph?

 

[Jan Hill]

I don't understand how that helps us get further with cosx = x having more than one solution

 

[Mark44]

My earlier question was: why do you think there are more than one solution? You said the the graphs of y = cosx and y = x intersect at three points. My response is that the only intersection point I find is when x is about .72.

I don't believe there are any other points of intersection. If you believe there are, what are the x values at these points?

 

[Jan Hill]

I initially looked for more than one solution for cosx = x because that's the way the question was posed but I see now that it was just to show that in fact, there is only one solution to it and that, as seen by looking at the graphs is x is about .72

Thank you for your help

 

[Mentallic]

The gradient of cosx is never more than 1 so a line of gradient 1 or more can only ever cut y=cosx once.

 

[Jan Hill]

Got it. thanks