Determine whether Rolle's Theorem can be applied to the function on the closed interval If Rolle's Theorem can be applied, find all numbers c in the open
Interval such that
A)Rolle's Theorem applies; c = 0
B)Rolle's Theorem applies;
C)Rolle's Theorem applies;
D)Rolle's Theorem applies;
E)Rolle's Theorem does not apply.

Question

Determine whether Rolle's Theorem can be applied to the function  on the closed interval  If Rolle's Theorem can be applied, find all numbers c in the open
Interval  such that 
A)Rolle's Theorem applies; c = 0 
B)Rolle's Theorem applies; 
C)Rolle's Theorem applies; 
D)Rolle's Theorem applies; 
E)Rolle's Theorem does not apply.

Quizplus · Accepted Answer

Rolle's Theorem can be applied if the function is continuous on the closed interval and differentiable on the open interval, and if the function values at the endpoints of the closed interval are equal. In this case, the function is continuous and differentiable on the given intervals, and f(-1) = f(1) = 0, so Rolle's Theorem applies. The only number c in the open interval (-1,1) where f'(c) = 0 is c = 0. Rolle's Theorem states that if a function $f(x)$ is continuous on a closed interval $[a,b]$, differentiable on the open interval $(a,b)$, and $f(a) = f(b)$, then there exists at least one number $c$ in $(a,b)$ such that $f'(c) = 0$.

In this case, the given function $f(x) = x^2 - 2x$ is continuous on $[0,2]$ and differentiable on $(0,2)$, and $f(0) = f(2) = 0$. Therefore, we can apply Rolle's Theorem to find a number $c$ in $(0,2)$ such that $f'(c) = 0$.

To find this number $c$, we need to find where the derivative $f'(x) = 2x - 2$ is equal to $0$ on the interval $(0,2)$. Setting $f'(x) = 0$, we get $2x - 2 = 0$, or $x = 1$. Therefore, the only number $c$ in $(0,2)$ where $f'(c) = 0$ is $c = 1$.

Thus, the answer is A, Rolle's Theorem applies; $c=0$.

Determine Whether Rolle's Theorem Can Be Applied to the Function