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:
 The function has to be continue in the interval
#[a,b]# ;  The function has to be derivable in the interval
#(a,b)# ; #f(a)=f(b)# .
If the hypotesis are satisfied, than there is a point

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:
and its domain is
#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!