what is the maximum number of relative extrema contained in the graph of this function
f(x)=3x^4-x^2+4x-2
again another question I'm not 100% on. my answer was 0

Question

what is the maximum number of relative extrema contained in the graph of this function
f(x)=3x^4-x^2+4x-2
again another question I'm not 100% on. my answer was 0

loser66 · Answer

nope

niuuu · Answer

@Loser66 what did I do wrong? I have no idea how to do it

loser66 · Answer

To find the relative extrema, you need to take derivative, right?

loser66 · Answer

and f'(x) = 12x^3-2x+4

loser66 · Answer

oh, I misread, the number of relative extrema is 0, you meant there is no relative extrema, right?

niuuu · Answer

yes sir

loser66 · Answer

To me, there is 1 extrema because f'(x) is a cubic, and the leading coefficient is 12 >0, it has 1 root

loser66 · Answer

--> f'(x) = 0 at somewhere --> 1 extrema. 
I calculated and got f'(x) =0 at x = -0.77317

mathmate · Answer

@niuuu 
For future reference, all real even-degree polynomials (except 0 degree) have at least one relative extremum.
As explained by @Loser66, we need to solve for the zero of the derivative, which is always an odd degree polynomial.  It turns out that all real odd-degree polynomials have at least one real root, hence at least one relative extremum.

mathmate · Answer

@Loser66 
The question is a little fishy.
If the student is not expected to solve a cubic, then the "maximum" number of relative extrema would be 3 (=4-1)!

loser66 · Answer

The Asker knew derivative, how can he not know how to solve a cubic?

loser66 · Answer

When using "relative extrema" notation, I believe that the problem is from calculus1. If it is not, I am sorry for my solution. :)

mathmate · Answer

I am sure it's from calculus.  Some programs can zip through calculus without knowing how to solve a cubic.  But the main suspect is the "maximum" number of maxima, which make me sound like student is not expected to solve the cubic.
In any case, your solution is perfect for the given problem.  I am not trying to split hairs!

mathmate · Answer

@loser66
I find the question ambiguous.
I would either say "What is the number of extrema....", or "What is the maximum number of possible extrema..." and would not say "What is the maximum number of extrema..."
So let's leave it at that.