Answer:
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:
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:
Furthermore, the length of the interval I is: . (See the graph)