Advanced Calculus: Proving x=cosx for x in (0,π/2)

[kimkibun]

Homework Statement

Show that x=cosx, for some xε(0,∏/2).

Homework Equations

The Attempt at a Solution

Define f(x)=x-cosx, i want to show that for some aε(0,∏/2), lim_x→af(x)=0. is this correct?

 

[Mentallic]

Homework Statement

Show that x=cosx, for some xε(0,∏/2).

Homework Equations

The Attempt at a Solution

Define f(x)=x-cosx, i want to show that for some aε(0,∏/2), lim_x→af(x)=0. is this correct?

No, that won't help you.
If the y value at x=0 is negative, and the y value at x = pi/2 is positive (these values can be shown because it's easy to compute them), then what can we conclude from this?

Does this logic extend to every function? Think about y=1/x, at x=-1 we have y=-1, and at x=1 we have y=1, but the function doesn't cross the x-axis at all.

 

[kimkibun]

No, that won't help you.
If the y value at x=0 is negative, and the y value at x = pi/2 is positive (these values can be shown because it's easy to compute them), then what can we conclude from this?

Does this logic extend to every function? Think about y=1/x, at x=-1 we have y=-1, and at x=1 we have y=1, but the function doesn't cross the x-axis at all.

do you have a better explanation sir?

 

[Mentallic]

do you have a better explanation sir?

I suppose.

Take the function y=2x. How do we show it crosses the x-axis between x=-1 and x=1?

Well, what is the y value at x=-1? y=2(-1)=-2. So at x=-1, the function is below the x-axis.
What about at x=1? y=2(1)=2, which is above the x-axis. So since the function went from below the x-axis at x=-1 to above the x-axis at x=1, does this mean we can conclude that it must've crossed the x-axis somewhere in between? Yes!

Why? Well again, think about the function y=1/x and try using the same procedure I just showed you. Everything seems to be the same, except that this function doesn't cross the x-axis. What's different?

 

[Borek]

It is a direct application of a known theorem - I guess it was discussed during lecture or is mentioned in your book.

Mentallic tries to guide you to the intuitive understanding behind this theorem.

 

[Mentallic]

It is a direct application of a known theorem - I guess it was discussed during lecture or is mentioned in your book.

Mentallic tries to guide you to the intuitive understanding behind this theorem.

Right, it was silly of me not to mention the theorem involved in solving this problem.

kimkibun, the Intermediate Value Theorem is what you're looking for.