1. Mar 19, 2013

### kimkibun

1. The problem statement, all variables and given/known data

Show that x=cosx, for some xε(0,∏/2).

2. Relevant equations

3. The attempt at a solution

Define f(x)=x-cosx, i want to show that for some aε(0,∏/2), limx→af(x)=0. is this correct?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Mar 19, 2013

### Mentallic

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.

3. Mar 19, 2013

### kimkibun

do you have a better explanation sir?

4. Mar 19, 2013

### Mentallic

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?

5. Mar 19, 2013

### Staff: Mentor

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.

6. Mar 19, 2013

### Mentallic

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.