Determine whether Rolle\'s theorem can be applied to f on the closed interval [a

Question

Determine whether Rolle's theorem can be applied to f on the closed interval [a,b]. If so find the values of c in the open interval (a,b) such that f 'c = 0. If rolle's theorem cannot be applied, explain why not.

Determine whether Rolle's Theorem can be applied to fon the closed interval [a,b] If so, find all values of c in the open interval (a,b) such that f (c)-0. If Rolle's If so, find all values of c in the open interval (a,b) such that J'(c)-0. If Rolle's Theorem cannot be applied, explain why not. , [0,1

Explanation / Answer

given f(x) = x(x-1)3 , [0,1]

f'x) = d/dx [x(x-1)3 ]

= (x-1)3 + x[2(x-1)2 ]

=  (x-1)3 + 2x(x-1)2

by rolls theorem f'(c)=0

  (c-1)3 + 2c(c-1)2 = 0

   (c-1)2[ c-1+ 2c] =0

  (c-1)2[ 3c-1] = 0

c-1 =0 and 3c-1=0

c=1 , c = 1/3  

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