O15-054
> restart;
> f:=(sin(x))/x;ff:=unapply(f,x);

                                  sin(x)
                             f := ------
                                    x


                                     sin(x)
                          ff := x -> ------
                                       x

> f2:=simplify(x^3*diff(f,x$2));

                              2
               f2 := -sin(x) x  - 2 cos(x) x + 2 sin(x)

> plot([f,f2],x=0..10,y=-20..20);

> assume(k,integer);
> v1:=simplify(subs(x=Pi/2+2*k*Pi,f2));v2:=simplify(subs(x=Pi/2+(2*k+1)*Pi,f2));

                            2          2       2   2
              v1 := - 1/4 Pi  - 2 k~ Pi  - 4 k~  Pi  + 2


                           2          2       2   2
               v2 := 9/4 Pi  + 6 k~ Pi  + 4 k~  Pi  - 2

> factor(v1*v2);# Ce qui est clairement négatif d'où l'existence d'au moins une racine

            2          2        2   2
  - 1/16 (Pi  + 8 k~ Pi  + 16 k~  Pi  - 8)

             2           2        2   2
        (9 Pi  + 24 k~ Pi  + 16 k~  Pi  - 8)


> tang:=u->y=ff(u)+(x-u)*D(ff)(u);

              tang := u -> y = ff(u) + (x - u) D(ff)(u)

> tang(u);

                    sin(u)           /cos(u)   sin(u)\
                y = ------ + (x - u) |------ - ------|
                      u              |  u         2  |
                                     \           u   /

> sys:=expand(subs(tang(u),x^2+4*y^2-4));# Equation aux "x" des points communs à la tangente et à l'ellipse

                       2                                 2
          2      sin(u)       x sin(u) cos(u)      sin(u)  x
  sys := x  + 16 ------- + 24 --------------- - 16 ---------
                    2                2                 3
                   u                u                 u

                                 2       2      2
              sin(u) cos(u)     x  cos(u)      x  cos(u) sin(u)
         - 16 ------------- + 4 ---------- - 8 ----------------
                    u                2                 3
                                    u                 u

                     2      2       2
             x cos(u)      x  sin(u)            2
         - 8 --------- + 4 ---------- + 4 cos(u)  - 4
                 u              4
                               u

> a:=[coeffs(sys,x)];
> 

                                     2
            sin(u) cos(u)      sin(u)                2
  a := [-16 ------------- + 16 ------- - 4 + 4 cos(u) ,
                  u               2
                                 u

                  2                              2
            sin(u)       sin(u) cos(u)     cos(u)
        -16 ------- + 24 ------------- - 8 -------,
               3               2              u
              u               u

                    2           2
              sin(u)      cos(u)      cos(u) sin(u)
        1 + 4 ------- + 4 ------- - 8 -------------]
                 4           2              3
                u           u              u

> discr:=simplify((a[2])^2-4*a[1]*a[3]);

                    3                  4           2  2
  discr := -16 (-4 u  sin(u) cos(u) - u  - 8 cos(u)  u

                                     2  4      2               2
         + 8 sin(u) cos(u) u + cos(u)  u  + 4 u  - 4 + 4 cos(u) )

           /  4
          /  u
         /

> num:=-u^4*discr/16;# Il reste à prouver que discr=0 lorsque f2=0

             3                  4           2  2
  num := -4 u  sin(u) cos(u) - u  - 8 cos(u)  u  + 8 sin(u) cos(u) u

                 2  4      2               2
         + cos(u)  u  + 4 u  - 4 + 4 cos(u)

> 
> num1:=simplify(subs({u^2=2-2*u*cos(u)/sin(u),u^3=2*u-2*u^2*cos(u)/sin(u),u^4=2*u^2-2*u^3*cos(u)/sin(u)},num));

                              2  2    2           3
  num1 := -2 (-5 sin(u) cos(u)  u  + u  sin(u) - u  cos(u)

                          2           3           3  3
         + 6 sin(u) cos(u)  - 8 cos(u)  u + cos(u)  u  - 2 sin(u)

         + 4 u cos(u))/sin(u)

> num2:=simplify(subs({u^2=2-2*u*cos(u)/sin(u),u^3=2*u-2*u^2*cos(u)/sin(u),u^4=2*u^2-2*u^3*cos(u)/sin(u)},num1));

                   2
  num2 := -4 cos(u)

                               2         2  2               2    /
        (-2 sin(u) cos(u) u - u  + cos(u)  u  + 2 - 2 cos(u) )  /  (
                                                               /

                   2
        -1 + cos(u) )

> num3:=simplify(subs({u^2=2-2*u*cos(u)/sin(u),u^3=2*u-2*u^2*cos(u)/sin(u),u^4=2*u^2-2*u^3*cos(u)/sin(u)},num2));# C'est gagné

                              num3 := 0


>