Matematické Fórum

Skrytý text:


we study equation   for ,  by the behaviour of the funcion

we have  , 



rewriting    as     shows that    if and only if   as function is increasing on the definition range
directly giving    and 

those observations tells us together with    that 
the function has a global minimum in   with  ,
and that   "grows" to continuously if moving to the left and also if moving to the right of

therefore there are exactly two points in definition range of such that

furthermore there is and hence  ,   giving that 
has in local maximum and in local minimum


now we have

  and   for    so for  reaching maximum at


then,   and     for    so from the positive maximum decreases to still positive value at   so no solutions for there

we can now notice that for   there is already    as   
and

thus the point   lies somewhere in the interval 

picking from the interval like (I picked closer to 1 because the value is obviously closer to 0 then
we get     that is obviously negative

so we learned that go negative on some subinterval of

together with what we learned about , and this tells us that it's here where crosses the line
for exactly two times and nowhere else it does

as is even function we get also 2 more negative solutions for which gives us total of 5