Verify Rolle's theorem for the function f(x) = x(x −2)2 on the interval [0, 2].
The given function is , which can be rewritten as .
We know that a polynomial function is everywhere derivable and hence continuous.
So, being a polynomial function, is continuous and derivable on .
Thus, all the conditions of Rolle's theorem are satisfied.
Now, we have to show that there exists such that .
Hence, Rolle's theorem is verified.
RD Sharma XII Vol 1 (2015)