A quartic is a polynomial of the 4th degree function. So the diffrential of the quartic is a third degree function f'(*x)
So , f'(x) = 0 gives all the critical points of extrema.
But a third degree equation can have at minimum one real root and a pair of complex conjugate roots or a maximum of all real three roots. So at minimum for a quartic there has to be one extrema.And at maximum it can have 3 extrema. Therefore, at minimum there could be one vertex for a quartic or at maximum it can have 3 vertices.