Hello friends,
I'm kinda stuck in the following problem. This is related to the derivative and possibly with the concept of maxima and minima. I've this assignment due on tuesday, and I am not getting any idea although trying it since two days before. can anybody help me, please?

For the cubic f(x) = ax3 + bx2 + cx + d, a ≠ 0, Find conditions on a, b, c, and d to ensure that f is always increasing or always decreasing on (-∞,+∞).

I also don't know where it comes from...sign is the factor that makes difference.
If a<0, must be: b>0 and c<0 [that ensures f is always decreasing]
If a>0, must be: b<0 and c>0 [that ensures f is always increasing]
It doesn't matter whether d is positive or negative or zero.
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since your assgnmt is due tuesday... I will try to find out more till then.

Ghamad,
Seems I did mistake..... I will write soon again...

f(x) = ax^3 + bx^2 + cx + d
f'(x) = 3ax^2 + 2bx + c

For the function to be either increasing or decreasing on the intervel (-inf, +inf), f'(x) = 0 should not have more than one root, because two roots mean there is a maxima and a minima, one root means there is one point of inflection, and no root means there is no mamima, minima, or point of inflection. So,

Required condition is,

discriminant of f'(x) <= 0; (0: one root, <0: no root)
or, (2b)^2 - 4(3a).c <= 0
or, b^2 <= 6ac

I hated it, sorry man can't be much of a help.

Though I donot know the right soln. I think Propensity is wrong.Propensity's staement are not general.

two roots mean there is a maxima and a minima...how abt if two roots are identical?
one root means there is one point of inflection...is perhaps true.

and no root means there is no mamima, minima...wat abt 2+0.1sin(60t)? this has no root but has max n minima.......

i might be wrong...as i m not a mathematics student.

Thanks for the try, propensity. but, are u sure that f ' (x) can be 0. If f '(x) is 0, then doesn't it mean that the function has some maximum or minimum value somewhere in its domain. Man, can u try this again please, coz I also need help in the same problem.

f'(x) can have two roots but there is no way to say which one is max and which one is
minimum. We have to assume that if one is max then other must be min.
Lets consider for root,
-b+sqrt(b^2-3ac)/3a
We take second derivative of f(x)
f'(x)=6ax+2b
substituting we get
f'(X)=2sqrt(b^2-3ac)
let 2sqrt(b^2-3ac)>0 so that f(x) is minimum at provided x
then to meet this condition
b^2>3ac, which is also the condition for the function to keep decreasing
(or increasing, cant say that yet) lets all think.

U guys are thinking too much.

well how does F(x)=A.X behave ( it is either increasing ORdecreasing From + infinity to - infinity) depending upon the sign of A

Similarly, How does F(X)=B.X^2 behave( its a parabola right so its increasing AND then decreasing or vice versa depending upon the sign of B)

Furthermore, how does F(x)=C.X^3 behave. It is also increaseing OR decreasing from + infinity to negative infinity depending upon the sign of C. This is becuase it will always have a infletion (ogee) point.

Now to the question. From the given information it should be obvious that the condition need to hold for A.x^3+B.X^2+C.X+D is

A+C-B>0 because this condition will over power how the X^2 behaves.

P.s D does not matter because it just shifts the intercept.

I think that is the condition needed for the function to be increasing or decreasing from negative infinity to positive infinity.

Best of Luck.

sorry I forgot one more condition..lol

so the condition needed is the function needs to behave so that the combination of A and C overpowers B.

one more condition needs to hold A>C

I think..lol..