Question

What are the values of $c$ for which Rolle's theorem for the function $f(x)=x^{3}-3 x^{2}+2 x$ in the interval $[0,2]$ is verified?

• A. $c=\pm 1$
• B. $c=1 \pm \frac{1}{\sqrt{3}}$
• C. $c=\pm 2$
• D. None of these

Answer 1

Answer: (B)

Here, we observe that
(a) $f(x)$ is a polynomial, so it is continuous in the interval $[0,2]$.
(b) $f'(x)=3 x^{2}-6 x+2$ exists for all $x \in(0,2)$.
So, $f(x)$ is differentiable for all $x \in(0,2)$ and
(c) $f(0)=0, f(2)=2^{3}-3(2)^{2}+2(2)=0$
$\Rightarrow f(0)=f(2)$
Thus, all the three conditions of Rolle's theorem are satisfied.
So, there must exist $c \in[0,2]$ such that $f'(c)=0$
$\Rightarrow f'(c)=3 c^{2}-6 c+2=0$
$\Rightarrow c=1 \pm \frac{1}{\sqrt{3}} \in[0,2]$