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

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.

& \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{aligned}$$ We can now see that the above derivative vanishes at two points $$\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.