Question

How‌ ‌to‌ ‌use‌ ‌Rolle’s‌ ‌theorem‌ ‌for‌ ‌\[f\left(‌ ‌x‌ ‌\right)=\left(‌ ‌\dfrac{{{x}^{3}}}{3}‌ ‌\right)-3x\]‌ ‌on‌ ‌the interval \[\left[ -3,3 \right]\]?

Answer 1

Hint: In this problem, we have to use Rolle’s theorem for the given function within the given interval. We know that Rolle’s theorem states that if a continuous differentiable function \[f\left( x \right)\] satisfies \[f\left( a \right)=f\left( b \right)=0,a< b\], then there is a point \[x\in \left( a,b \right)\] where \[f'\left( x \right)\] vanishes. We have to see whether the conditions in the Rolle’s theorem are satisfied as we use Rolle’s theorem for the given function.

Complete step-by-step solution:
We know that the given function is,
\[f\left( x \right)=\left( \dfrac{{{x}^{3}}}{3} \right)-3x\] on the interval \[\left[ -3,3 \right]\].
We can now see that,
\[\Rightarrow f\left( x \right)\] is a continuous function.
\[\Rightarrow f\left( -3 \right)=0=f\left( 3 \right)\]
When we apply the given values of a and b in \[f\left( x \right)\], we get 0.
Thus, the conditions of Rolle’s theorem are satisfied with a = -3 and b = 3 and so there is a \[x\in \left( a,b \right)\]which satisfies \[f'\left( x \right)=0\].
We can now find \[f'\left( x \right)\]by differentiating \[f\left( x \right)\].
\[\Rightarrow f'\left( x \right)={{x}^{2}}-3\]
We can now substitute \[\pm \sqrt{3}\] in \[f'\left( x \right)\], to get the condition to be satisfied.
\[\begin{align}
  & \Rightarrow f'\left( x \right)={{\left( \sqrt{3} \right)}^{2}}-3=3-3=0 \\
 & \Rightarrow f'\left( x \right)={{\left( -\sqrt{3} \right)}^{2}}-3=3-3=0 \\
\end{align}\]
 We can now see that the above derivative vanishes at two points \[\pm \sqrt{3}\] in the interval \[\left( -3,3 \right)\].
Therefore, Rolle’s theorem is satisfied for the given function \[f\left( x \right)=\left( \dfrac{{{x}^{3}}}{3} \right)-3x\] on the interval \[\left[ -3,3 \right]\].

Note: We should remember that Rolle’s theorem says that there is a \[x\in \left( a,b \right)\] where \[f'\left( x \right)\] will vanish, not that there will necessarily be only one. We should also know that the given function should be a continuous differentiable function, which can be further analysed for the conditions in Rolle's theorem.