$\text { Determine whether Rolle's Theorem can be applied to } f ( x ) = - x ^ { 2 } + 10 x \text { on the closed }$ interval $[ 0,10 ]$. If Rolle's Theorem can be applied, find all values of $c$ in the open interval $( 0,10 )$ such that $f ^ { t } ( c ) = 0$
A) Rolle's Theorem applies; $c = 6 , c = 5$
B) Rolle's Theorem applies; $c = 4 , c = 6$
C) Rolle's Theorem applies; $c = 5$
D) Rolle's Theorem applies; $c = 4$
E) Rolle's Theorem does not apply

Question

Quizplus · Accepted Answer

For Rolle's Theorem to apply, the function must be continuous on the closed interval and differentiable on the open interval. 
Here, $f(x)=-x^2+10x$ is a polynomial function and is continuous everywhere. Also, $f(x)$ is differentiable on $(0,10)$ because its derivative, $f'(x)=-2x+10$, is defined and continuous on the open interval. 
Therefore, Rolle's Theorem applies.

Next, we need to find the values of $c$ in $(0,10)$ such that $f'(c)=0$. 
Since $f'(x)=-2x+10$, we have:

$$f'(c)=0 \iff -2c+10=0 \iff c=5$$

Therefore, the only value of $c$ that satisfies the conditions of Rolle's Theorem is $c=5$, which is the answer choice C.

Determine whether Rolle’s Theorem can be applied to f(x)=−x2+10x on the closed \text { Determine whether Rolle's Theorem can be applied to } f ( x ) = - x ^ { 2 } + 10 x \text { on the closed } Determine whether Rolle’s Theorem can be applied to f(x)=−x2+10x on the closed

$\text { Determine whether Rolle's Theorem can be applied to } f ( x ) = - x ^ { 2 } + 10 x \text { on the closed }$