Still have math questions?

Ask our expert tutors
Algebra
Question

Decide if Rolle's Theorem can be applied to \( f (...

Decide if Rolle's Theorem can be applied to \( f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 4 x ^ { 2 } + 1 \) on the interval \( [ - 1,3 ] \) . If Rolle's Theorem can be applied, find all values \( c \) in the interval such that \( f ^ { \prime } ( c ) = 0 \) . If Rolle's Theorem cannot be applied, state why. 

Answer

Your input: find all numbers \( c \) (with steps shown) to satisfy the conclusions of the Mean Value Theorem for the function \( f = x ^ { 4 } - 4 x ^ { 3 } + 4 x ^ { 2 } + 1 \) on the interval \( [ - 1,3 ] \) . 

The Mean Value Theorem states that for a continuous and differentiable function \( f ( x ) \) on the interval \( [ a , b ] \) there exists such number \( c \) from the interval \( ( a , b ) \) , that \( f ^ { \prime } ( c ) = \frac { f ( b ) - f ( a ) } { b - a } \) 

First, evaluate the function at the endpoints of the interval: 

\(f ( 3 ) = 10 \) 

\( f ( - 1 ) = 10\) 

Next, find the derivative: \( f ^ { \prime } ( c ) = 4 c ^ { 3 } - 12 c ^ { 2 } + 8 c\) 

Form the equation: \( 4 c ^ { 3 } - 12 c ^ { 2 } + 8 c = \frac { ( 10 ) - ( 10 ) } { ( 3 ) - ( - 1 ) } \) 

Simplify: \( 4 c ^ { 3 } - 12 c ^ { 2 } + 8 c = 0\) 

Solve the equation on the given interval: \( c = 0 , c = 1 , c = 2\) 

Answer: \( 0,1,2\) 

Solution
View full explanation on CameraMath App.