Homework Help: General 3rd degree polynomial always increasing problem? Please Help

1. Oct 16, 2008

arpitm08

General 3rd degree polynomial always increasing problem? Please Help!!!

What conditions on a, b, and c will make f(x)=ax^3+bx^2+cx+d always increasing?

For a function to be always increasing, the first derivative has to be always positive. So,

f1(x)=3ax^2+2bx+c>0

I tried finding the roots, that didn't lead me anywhere. Could someone please help?

I know that for a 3rd degree polynomial to be always increasing, it has to be a perfect cube. like (ex-f)^3, then you can expand that to be (ex)^3 -3f(ex)^2+3exf^2+f^3. Then since this will have to be the formula for f(x). So a=e^3, b=-3fe^2, c=3ef^2, d=f^3, but i don't know what to do from here. Please help!!!

2. Oct 17, 2008

Staff: Mentor

Re: General 3rd degree polynomial always increasing problem? Please Help!!!

I think I finally figured out what you meant by (ex-f)^3. It would have been easier to grasp if you had written (Ax - B)^3, since e and f already have other meanings -- the natural number e, and f as in f(x).

Some things to think about.
a should be positive.
There should be exactly 1 real root. (If there were 3 roots, the graph would have something of an S shape. If there were 2 roots, one root would be repeated, and the graph would drop down and touch the x axis rather than cross it.)
See if you can make up equations for 3rd degree polynomials with 1 root, 2 roots, 3 roots.
f'(x) should be >= 0 for all x.

What about f''(x)? Can f''(x) change sign? If so, the concavity is changing. If so, how many times can the concavity change sign?

3. Oct 17, 2008

HallsofIvy

Re: General 3rd degree polynomial always increasing problem? Please Help!!!

I wish you had shown what you did. The derivative is a quadratic so will always be positive if and only if a is positive and the quadratic is never 0. That will happen when the discrimant is negative: that is if (2b)2- 4(3a)(c)= 4b2- 12ac< 0 which is the same as b2< 3ac.

The cubic f(x)= ax3+ bx2+ cx+ d is always increasing if and only if a> 0 and b2< 3ac. It is always decreasing if and only if a< 0 and b2< 3ac.

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