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)) $.