Draw a function that is continuous on [0,1] with f(0) = 0, f(1) = 1, and f(0.5) = 20.
There are infinitely many right answers to this. Here's one of them:
We could change the above problem to make f(0.5) equal anything we want. Therefore
Be Careful: We can never use the IVT to conclude that a function ffails to hit a value M.
Another thing to be aware of with the IVT is that it doesn't tell us where a function hits a value M, or how many times it does so. On its best day, the IVT can only guarantee we hit the value M at least once.