Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. cos x = x. How do I begin this problem?

According to the theorem, a=0, b=1 and N=x?

To use the Intermediate Value Theorem to show that there is a root of the equation cos(x) = x in the interval [a, b], we need to ensure that the function cos(x) is continuous on the interval [a, b]. In this case, the interval [a, b] is [0, 1].

To begin solving this problem, we can start by verifying that the function cos(x) is indeed continuous on the interval [0, 1]. The cosine function, cos(x), is continuous for all real numbers, which means it is continuous on the interval [0, 1].

Now that we have established the continuity of the function on the given interval, we can proceed to evaluate the function at the endpoints of the interval.

Evaluate cos(0): The cosine of 0 is equal to 1, so cos(0) = 1.

Evaluate cos(1): The cosine of 1 is approximately 0.5403, so cos(1) ≈ 0.5403.

Since cos(0) = 1 and cos(1) ≈ 0.5403, the values of the function at the endpoints are not equal (1 ≠ 0.5403).

According to the Intermediate Value Theorem, if a function is continuous on a closed interval [a, b], and a value M lies in between f(a) and f(b), then there exists a value c in the interval (a, b) such that f(c) = M.

In this case, since f(0) = 1 and f(1) ≈ 0.5403, and 0 lies between them, we can conclude that there exists a value c in the interval (0, 1) such that f(c) = 0.

Therefore, the equation cos(x) = x has a root in the interval [0, 1].