Question

For the general cubic polynomial f(x)=ax3+bx2+cx+d (a is not equal to 0), find conditions for a, b, c, and d to assure that f(x) is always increasing or decreasing on (-infintiy,+infinity). Also prove that f(x) has exactly one inflection point.

Answer 1

Have you considered a 1st Derivative?

Answer 2

yes f'(x)=3ax2+2bx+c

Answer 3

Perfect. How do you ensure that this thing is ALWAYS positive?

Answer 4

It has to be above the x-axis or something like that right?

Answer 5

Yes. In completing the square, you start with f'(x) = 3a(x^2 + [(2b)/(3a)] + ___) - 3a(____). I think we just established that a > 0.

Answer 6

Okand for it to always be negative a<0?

Answer 7

Well, that's just a start. We don't have the whole answer, just yet.

For one thing, we have not considered 'd' at all. gtg. Let's see what you get.

Answer 8

For ensuring strict monotony, we require \(f'>0\) or \(f'<0\).$$f'(x)=3ax^2+2bx+c$$We also know that inflection points occur where \(f''=0\) -- show this only happens once:$$f''(x)=6ax+2b\\0=6ax+2b\\0=3ax+b\\3ax=-b\\x=-\frac{b}{3a}$$