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((:detse:discrime DISCRIMINANT))
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==Problems==
1.
Find the parameter value $ {\color{Red} \alpha } \in [-5,5] $ for which
$$ \max_{x\in]-\infty,+\infty[} (-x^6+{\color{Red} \alpha } \,x^5+3\,x^3+5\,x^2-2\,x-1) $$
is minimal. Evaluate this minimum.
2.
Establish the validity or invalidity of the following equality
$$ \mathcal D_x(\mathcal D_y (f(x,y))=\mathcal D_y(\mathcal D_x (f(x,y)) \, . $$
Here $ \mathcal D_x $ and $ \mathcal D_y $ denotes the discriminants of the polynomials treated with respect to the indexed variables. Compute
$$ \gcd(\mathcal D_x(\mathcal D_y (f(x,y)),\mathcal D_y(\mathcal D_x (f(x,y))) \quad \mbox{ for } \quad f(x,y)= -x^4-2\,y^4+4\,x^2-{\color{Red} \alpha } \,y^2+3\,xy+4\,y \ . $$
3.
Prove that the discriminan of the polynomial $ f(x)=x^4+p\,x^2+q\,x+r $ coincides with the discriminant
of its Ferrari's resolvent
$$ \mathcal D_x(f(x))= \mathcal D_t( t^3-p\,t^2-4\,r\,t+(4\,pr-q^2)) \ . $$
4.
Prove that if the polynomial of the degree $ 4 $ has its discriminant positve and two of its roots real, then all its roots are real.
5.
Prove that $ \mathcal D (f(x)f^{\prime \prime}(x)- [f^{\prime}(x)]^2) $,
treated as a polynomial with respect to the coefficients of $ f(x) $, is divisible by $ \mathcal D (f(x)) $.