Question

# Verify Rolle's theorem for the function f(x) = x(x −2)2 on the interval [0, 2].

Solution

## The given function is $f\left(x\right)=x{\left(x-2\right)}^{2}$, which can be rewritten as $f\left(x\right)={x}^{3}-4{x}^{2}+4x$. We know that a polynomial function is everywhere derivable and hence continuous. So, being a polynomial function, $f\left(x\right)$ is continuous and derivable on .  Also,  $f\left(0\right)=f\left(2\right)=0$ Thus, all the conditions of Rolle's theorem are satisfied. Now, we have to show that there exists  such that $f\text{'}\left(c\right)=0$. We have Thus, . Hence, Rolle's theorem is verified.MathematicsRD Sharma XII Vol 1 (2015)Standard XII

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