Question

This is an example in my text book but i don't understand how its being worked out........"We use Rolle's Theorem to show that p(x)=2X^3+5X-1 has exactly one real zero........... HOW DO I SHOW THIS?????

Answer 1

what is the derivative?

Answer 2

6x^2+5... Right?

Answer 3

yes that is correct
does the derivative have any real zeros?

Answer 4

?

Answer 5

i hope it is more or less clear that the derivative is never zero, because \(6x^2\geq 0\) for all \(x\) and so \(6x^2+5\geq 5\) and since it is always larger than 5, it is never zero

Answer 6

this means
a) the function is always increasing, because the derivative is always positive
if it is always increasing it cannot be zero twice
b) by rolles theorem, if the function had two zeros, then the derivative would have to be equal to zero somewhere between those two zeros
but the derivative is never zero, so the function cannot have two zeros

Answer 7

Srry i needed to go bak to the basics. Is THe derivative never zero because the limit is alwayys approaching zero?