Question

Verify the Rolle's theorem for the function $f\left(x\right)=x^2-3x+2$ on the interval[1,2]

• A. No Rolle's theorem is not applicable in the given interval
• B. Yes Rolle's theorem is applicable in the given interval and the stationary point $x=\frac{5}{4}$
• C. Yes Rolle's theorem is applicable in the given interval and the stationary point $x=\frac{3}{2}$
• D. nnone of these

Answer 1

Answer: (C)

Yes Rolle's theorem is applicable in the given interval and the stationary point $x=\frac{3}{2}$

It can be easily seen that $f\left(x\right)=x^2-3x+2$ is continuous as differentiable on R (being a polynomial) $\Rightarrow f\left(x\right)$ is continous in (1,2) and differentiable in [1,2]. Also, we have
$f\left(1\right)=f\left(2\right)=0$.

Thus, $f\left(x\right)$ satisfies all the conditions of Rolle's theorem in $[1,2]$ $\Rightarrow \exists$ at least one number, say $x$ in $[1,2]$ such that $f^{{}^{\prime }}\left(c\right)=0.$ Now, $f^{{}^{\prime }}\left(x\right)=2x-3=0\Rightarrow x=\frac{3}{2}$ Since, the root (stationary point) $x=\frac{3}{2}$ lies in the interval(1,2).

Hence Rolle's theorem is verified.