Question

Verify Rolle's theorem the function $f\left(x\right)=x^3-4x$ on $[-2,2].$ If you think it is applicable in the given interval then find the stationary point ?

• A. Yes Rolle's theorem is applicable and stationary point is $x=\pm \frac{2}{\sqrt{3}}$
• B. No Rolle's theorem is not applicable
• C. Yes Rolle's theorem is applicable and $x=2or-2$
• D. none of these

Answer 1

Answer: (A)

Yes Rolle's theorem is applicable and stationary point is $x=\pm \frac{2}{\sqrt{3}}$

The function $f\left(x\right)=x^3-4x$ is a polynomial and so it is continuous and differentiable at all x$ϵ$ R.
In particular it is continuous in the closed interval $[-2,2]$ .
Also $f(-2)]=0=f(2)$. Thus , $f\left(x\right)$ satisfies all three conditions of Rolle's theorem in $(-2,2)$.
Therefore , there must exist at least one real number ${}^{\prime }{x}^{\prime }$ in the
open interval $(-2,2)$ for which $f^{\prime }\left(x\right)=0$
Also $f^{\prime }\left(x\right)=3x^2-4$
Now $f^{{}^{\prime }}\left(x\right)=0$ gives $3x^2-4=0$ or $x=\pm \frac{2}{\sqrt{3}}$ which is also known as stationary point.
Both these value lie in the open interval $(-2,2)$ and thus the conclusion of Rolle's theorem is verified.