Number of real roots

[InterserveVB]

How can I find out how many real roots an equation has by using either rolle's theorem or the mean value theorem?

For example, x^5+5x+1 = 0

 

[pka]

The derivative is f’(x)=5x<SUP>4</SUP>+5. It is positive for all x.
What does that say about f, is it increasing?
Graph the function. There is a root between −1/2 and 0.