Question

What happens when the signs of the function in IVT are the same? Is it true that they don’t have any roots? Show an example

Answer 1

Yes, if the function has same sign, i.e., it is always positive or negative on your domain interval, the function has no root. But, note that IVT depends on the domain too.

For example, you can consider simple function f(x)=x on [2,3]. Then, your function value is always positive, because in the interval [2,3] your function f(x)=x takes value is between 2 and 3. But, if you take your domain to be [-1,1], then f(x)=x will have a root at x=0.

In general, you can look at the IVT geometrically as follows:

An understanding of the theorem statement is drawn down.

Now, the function value has plated in a graph. Here we have image of a under f is positive and the image of b under f is negative. I.e., f(a)> 0 and f(b)<0. Hence, from the following graph you can see that to draw a graph continuous from the point (a,f(a)) to (b,f(b)) without any breaking you have to cross x-axis. That is where your function value is 0. Hence, there is a root for the function f.