# Thread: Show that the largest positive root of the equaion x^3-3x^2-2=0 lies between 3 and 4.

1. ## Show that the largest positive root of the equaion x^3-3x^2-2=0 lies between 3 and 4.

Show that the largest positive root of the equaion x^3-3x^2-2=0 lies between 3 and 4.
How do I solve the question? I have no formula to use except trial and error method to find the first root, and then solve the quadratic equation. So how can i tell where the largest positive root of the equation lies?
I think it has something to do with f(3) and f(4), f being the cubic equation's expression, but I just cannot think out the exact procedure.

I have to use a sequence of linear interpolations to estimate this root correct to 2 decimal places later in the same question, so if I am allowed find out the root by trial and error and the quadratic equation it doesn't make any sense to have the second question.
I am really

2. [This is really a Calculus question, it doesn't belong in the Algebra section.]

If $f(x) = x^3-3x^2-2$ then f(3) and f(4) have different signs, so f must have a zero somewhere between 3 and 4 (that's because of the intermediate value theorem, which you probably ought to know).

Also, f(4)>0, and if you differentiate f you'll find that the derivative is positive whenever x>4. So f(x) is increasing for x>4 and so it can never decrease down to 0 again.