Question

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

Answer 1

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!