How do you verify that the hypotheses of rolles theorem are right for #f(x)= x sqrt(x+2)# over the interval [2,4]?

1 Answer
Apr 13, 2015

The hypotesis of the Rolles theorem are:

  1. The function has to be continue in the interval #[a,b]#;
  2. The function has to be derivable in the interval #(a,b)#;
  3. #f(a)=f(b)#.

If the hypotesis are satisfied, than there is a point #P(c,f(c))# such as #a < b < c# in which:

#f'(c)=0#.

  • Our function has domain #[-2,+oo]# (the radicand has to be positive or zero): so the function is continue in #[2,4]#.

  • The derivative is:

#y'=1*sqrt(x+2)+x*1/(2sqrt(x+2))#

and its domain is #(-2,+oo)#, because the root has gone at the dominator so the radicand has to be only positive: so the functionis derivable in #(2,4)#.

  • #f(2)=2*sqrt(2+2)=4# and #f(4)=4*sqrt(4+2)=4sqrt6# and they are different!

So the theorem can't be applicated!