# How do you verify whether rolle's theorem can be applied to the function f(x)=x^3 in [1,3]?

Oct 9, 2016

When we are asked whether some theorem "can be applied" to some situation, we are really being asked "Are the hypotheses of the theorem true for this situation?"

#### Explanation:

(The hypotheses are also called the antecedent, of 'the if parts'.)

So we need to determine whether the hypotheses of Rolle's Theorem are true for the function

$f \left(x\right) = {x}^{3}$ on the interval $\left[1 , 3\right]$

Rolle's Theorem has three hypotheses:

H1 : $f$ is continuous on the closed interval $\left[a , b\right]$
H2 : $f$ is differentiable on the open interval $\left(a , b\right)$.
H3 : $f \left(a\right) = f \left(b\right)$

We say that we can apply Rolle's Theorem if all 3 hypotheses are true.

H1 : The function $f$ in this problem is continuous on $\left[1 , 3\right]$ [Because, this function is a polynomial so it is continuous at every real number.]

H2 : The function $f$ in this problem is differentiable on $\left(1 , 3\right)$
[Because the derivative, $f ' \left(x\right) = 3 {x}^{2}$ exists for all real $x$. In particular, it exists for all $x$ in $\left(1 , 3\right)$.)

H3 : $f \left(1\right) = {1}^{3} = 1$ and $f \left(3\right) = {3}^{3} = 27$.
No, we do not have $f \left(a\right) = f \left(b\right)$. The third hypothesis is false for this function on this interval.

Therefore we cannot apply Rolle's Theorem to $f \left(x\right) = {x}^{3}$ on the interval $\left[1 , 3\right]$. (Meaning "at least one hypothesis is false".).)