quartic functions that have "W" shape?

Question

Can anyone help me and state conditions that must be fulfilled in order for quartic f. to have "W" (or M ) shape

Answer 1

this is simple
if we get f ' ( x ) where f ' ( x ) will be a cubic function
and we put f'(x) =0
if f ' (x) contains 3 different real roots , so quartic function f(x) will be M or W shaped

Answer 2

If your quartic is ax^4+bx^3+cx^2+dx+e then it will have a W-ish (or M) shape (as opposed to a U shape) if:
27b³d - 9b²c² + 32ac³ - 108abcd + 108a²d² > 0

(Ask a difficult question get a difficult answer)

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To explain why is fairly complicated so this is really only here for those other who may be interested.
The function above will have 3 local maxima and minima only if the derivative 4ax³+3bx²+2cx+d has three real roots. By completing the cubic it is possible to solve this and al I have done is sub in 4a,3b,2c,d into the cubic determinant then divided by a common factor..
More info in the link below

Answer 3

Basically, that's one of two possibilities, where the other possibility is that the first derivative is monotonic (in which case you'd have more of a U shape).

So, as stated above, you want the cubic polynomial that is the first derivative to have three real roots rather than 1 (those are the two possibilities).

I don't think you can get an answer any more elementary than that, because I don't think there's a simple test to count the number of real roots of a cubic polynomial.

Answer 4

W Shaped Graph

Answer 5

The leading coefficient is positive and f'(x) = 0 has three different real roots.

Answer 6

It is called a parabola, and The coefficient of x is positive and is expressed as f(x) = 0. x must be to the power of 2 or greater.

Answer 7

must not be a one - one function or like you can say one value of y many values of x.