for what range of values of k does the equation 12x+3x^2-2x^3=k have  a) Exactly three real roots? b) Two real roots? c) Only one real root?

Expert Answers

An illustration of the letter 'A' in a speech bubbles

We need to find where the turning points are of the cubic on the lefthand side of the equation. We do this by differentiating the cubic with respect to `x`:

`d/(dx) (12x + 3x^2 - 2x^3) = 12 + 6x - 6x^2`

To find the turning points we solve

`-6x^2 + 6x + 12 = 0`

ie, `x^2 - x - 2 = 0`

By insepction

`(x+1)(x-2) = 0`

So the turning points are at `x = -1` and `x = 2`

When `k` is between...

(The entire section contains 2 answers and 206 words.)

Unlock This Answer Now

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Start your 48-Hour Free Trial
Approved by eNotes Editorial Team

Posted on

An illustration of the letter 'A' in a speech bubbles
Approved by eNotes Editorial Team

Posted on