Question

f,g:[1,2]->[1,2] continue astfel incat g (1)=1, g (2)=2. Demonstrati ca f (x)=g (x) are solutii in [1,2].

Answer 1

1<=f(x)<=2;   1<=g(x)<=2; pentru orice x din [1,2]
Fie h:[1,2]->R,h(x)=f(x)-g(x), h continua
h(1)=f(1)-g(1)=f(1)-1
1<=f(1)<=2   0<=f(1)-1<=1........h(1)>=0
h(2)=f(2)-g(2)=f(2)-2
1<=f(2)<=2   -1<=f(2)-2<=0   h(2)<=0

h continua pe [1,2](I)
h(1)*h(2)<=0(II)

Din (I) si (II) rezulta ca ecuatia h(x)=0 are cel putin o solutie in [1,2]
deci ecuatia f(x)=g(x) are solutii in [1,2]