What Coefficients Create a W Shape in a Quartic Function?

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The discussion focuses on the conditions necessary for a quartic function, represented by the equation y=Ax^4+Bx^3+Cx^2+Dx+E, to exhibit a "W" shape. It is established that the coefficient A must be greater than zero (A > 0) and that the constant term E can take any value. Additionally, the function must possess two distinct minima and one maximum, which implies that the derivative can be expressed as a product of three linear terms: a(x-u)(x-v)(x-w), where u, v, and w are distinct real numbers.

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Quartic function Problem! EMS!

I got stuck on a question that tells me to find the restriction on coefficient on equation y=Ax^4+Bx^3+Cx^2+Dx+E, so that the function can form a "w" shape, and I found that a>0 and e can be any thing, but others I cannot find. SO if anyone PLZ help me
 
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In order to have a "w" shape, the function must have two distinct minima and a maximum- that is, the derivative can be written as a product of three linear terms: a(x- u)(x- v)(x- w) with u, v, w distinct numbers.
 

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