Question

determine whether rolles theorem can be applied to f on the closed interval a, b. (select all that apply.)
\\( f ( x ) = - x ^ { 2 } + 2 x, \quad 0,2 \\)
yes, rolles theorem can be applied.
no, because f is not continuous on the closed interval a, b.
no, because f is not differentiable on the open interval (a, b).
no, because \\( f ( a ) \
eq f ( b ) \\).
if rolles theorem can be applied, find all values of c in the open interval (a, b) such that \\( f ^ { \prime } ( c ) = 0 \\). (enter your answers as a comma - separated list. if rolles theorem cannot be applied, enter na.)
c =

Explanation:

Step1: Check continuity and differentiability

Since \(f(x)=-x^{2}+2x\) is a polynomial function, it is continuous on \([0,2]\) and differentiable on \((0,2)\).

Step2: Check \(f(a) = f(b)\)

Calculate \(f(0)=-(0)^{2}+2(0)=0\) and \(f(2)=-(2)^{2}+2(2)= - 4 + 4=0\). So \(f(0)=f(2)\).

Step3: Find \(f^{\prime}(x)\)

Differentiate \(f(x)\) using the power rule. If \(y = ax^{n}\), then \(y^{\prime}=nax^{n - 1}\). For \(f(x)=-x^{2}+2x\), \(f^{\prime}(x)=-2x + 2\).

Step4: Solve \(f^{\prime}(c)=0\)

Set \(f^{\prime}(c)=-2c + 2 = 0\).

$$
LATEXBLOCK0
$$

Answer:

\(1\)