Suppose we have x3+αx+β=0
Problem:
the cubic equation has one real root if α>0 and three real roots if 4α3+27β2<0.
Try:
Let f(x)=x3+αx+β. One has that f′(x)=3x2+α. We have critical points when
3x2+α=0⟺x2=−α3
Clearly, if α>0, then we dont have critical points and f′(x)>0 thus always increasing. Since lim and \lim_{x \to -\infty} = - \infty, then there must be some c such that f(c) = 0, so we have one real root.
Now, if \alpha < 0, then we have critical points x = \pm \sqrt{ -\alpha/3 } . but, here I am stuck. Any help would be greatly appreaciated.
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