determine whether rolles theorem can be applied to f on the closed interval a, b. (select all that…

determine whether rolles theorem can be applied to f on the closed interval a, b. (select all that apply.)\n\\( f ( x ) = - x ^ { 2 } + 2 x, \\quad 0,2 \\)\nyes, rolles theorem can be applied.\nno, because f is not continuous on the closed interval a, b.\nno, because f is not differentiable on the open interval (a, b).\nno, because \\( f ( a ) \\neq f ( b ) \\).\nif 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.)\nc =

determine whether rolles theorem can be applied to f on the closed interval a, b. (select all that apply.)\n\\( f ( x ) = - x ^ { 2 } + 2 x, \\quad 0,2 \\)\nyes, rolles theorem can be applied.\nno, because f is not continuous on the closed interval a, b.\nno, because f is not differentiable on the open interval (a, b).\nno, because \\( f ( a ) \\neq f ( b ) \\).\nif 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.)\nc =

Answer

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). [ \begin{align*} -2c+2&=0\ -2c&=-2\ c& = 1 \end{align*} ]

Answer:

(1)