Show that cosx=x has a solution in the interval [0,1] . Hint: Show that f(x)=x-cosx has a zero i

Show that cosx=x has a solution in the interval [0,1] ....

Key Concepts

Continuous Functions

A function is continuous if small changes in the input result in small changes in the output, meaning there are no abrupt jumps or breaks in the graph. In the context of this problem, the cosine function and the linear function are both continuous, which makes the combination f(x) = x - cos x continuous over the interval [0, 1]. This property is essential because it ensures that all intermediate values between f(0) and f(1) are attained, a condition needed for applying the Intermediate Value Theorem.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes different signs at the endpoints, then there exists at least one point c in the interval where the function equals zero. In this problem, by showing that f(0) and f(1) have opposite signs for f(x) = x - cos x, one can conclude that f must cross zero at some point between 0 and 1, thereby proving that cos x = x for some x in [0, 1].

Transcript

00:01 We will show that the equation x equal cosine of x has a solution at the interval 0 1.

00:10 So first we're going to give some geometric insight to this equality, to this equation.

00:20 We are going to draw the graph of the cosine functions.

00:28 We had something like this, maybe a little bit better, something like this, where this point here, is by half and this value here is one when x equals zero so um we have this function and then put the here and then we have the identity that is let's put this way this is cosine of x and now we draw the identity y equals x and we know it's this way something like this so the solution of this equation or any x that is a solution of this equation corresponds to the intersection of the two graph so we're talking about this point here and as we can expect from the figure there is only one point where that occur at least between 0 and by half as we see here but that's a graphic inside only so we get to do some calculations here to verify that there is a solution to this equation between 0 and 1 so we know that by half is about 1 .5 or something like that so number 1 is somewhere where we guess from here.

02:29 So we want to show that in fact, if the number one is here, the solution of the equation is to the left of one and is positive.

02:41 So we are going to see that in a moment.

02:44 So we are going to define the function f of x equal x minus cosine of x for x on the interval 0 1.

02:58 And we are going to see what happens.

03:01 To the function on the endpoints of the interval.

03:05 Before that, we know that the main property of this function is that f is continuous on 01, because x is continuous and cosine of x is continuous everywhere, just like the identity x.

03:27 So in particular, over the interval 01, both functions are continuous, and so there are difference...

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