How do you determine if rolles theorem can be applied to #f(x) = 2 − 20x + 2x^2# on the interval [4,6] and if so how do you find all the values of c in the interval for which f'(c)=0?

1 Answer
Jun 22, 2015

It is possible to apply and the answer is #c=5#.

Explanation:

The Rolles theorem says that if:

  1. #y=f(x)# is a continue function in a set #[a,b]#;
  2. #y=f(x)# is a derivable function in a set #(a,b)#;
  3. #f(a)=f(b)#;

then at least one #cin(a,b)# as if #f'(c)=0# exists.

So:

  1. #y=2-20x+2x^2# is a function that is continue in all #RR#, and so it is in #[4,6]#;
  2. #y'=-20+4x# is a function continue in all #RR#, so our function is derivable in all #RR#, so it is in #[4,6]#;
  3. #f(4)=f(6)=-46#.

To find #c#, we have to solve:

#y'(c)=0rArr-20+4c=0rArr4c=20rArrc=5#

(the value is #in[4,6]#).