Answer:
![I = [0.86,0.87]](https://tex.z-dn.net/?f=I%20%3D%20%5B0.86%2C0.87%5D)
Step-by-step explanation:
The Intermediatre Value Theorem (IVT) states the following:
If f is a continuous function on the interval [a,b], then f assumes any value between f(a) and f(b) at some point within the interval, i.e., for every z in the image of f, there is w in [a,b] such that z = f(w).
An important consequence of this result is that if the continuous function f has values of opposite sign inside the interval [a,b], then f has a root in that interval, i.e., there is w in [a,b] satisfying f(w)=0.
We are going to show that the function f defined by
has a real root by seeing that f changes its sign on the interval [0, 1]. In fact:
![f(1)=\cos(1)-1^3\approx -0.46](https://tex.z-dn.net/?f=f%281%29%3D%5Ccos%281%29-1%5E3%5Capprox%20-0.46)
Then, as a consequence of the IVT, there is w in [0,1] such that f(w)=0, or equivalently,
, as desired.
Now, doing some calculations we found that the interval
contains the root w of f, since f changes its sign here:
![f(0.86)=cos(0.86)-(0.86)^3\approx 0.02](https://tex.z-dn.net/?f=f%280.86%29%3Dcos%280.86%29-%280.86%29%5E3%5Capprox%200.02)
![f(0.87)=cos(0.87)-(0.87)^3\approx -0.01](https://tex.z-dn.net/?f=f%280.87%29%3Dcos%280.87%29-%280.87%29%5E3%5Capprox%20-0.01)
Furthermore, the length of the interval I is:
. (See the graph)