There is one function f:[-1,1]->R, defined like this:
f(x)=x*x+m*x+n for x in [-1,0) and
f(x)=p*x*x+4*x+4 for x in [0,1] and m,n,p in R. To be find the parameters m, n, p so as the function f verify the conditions of theoreme's Rolle on the interval [-1,1].

For f to be in conditions of theorem's Rolle she must be continue on [-1,1], derivate on (-1,1) and f(-1)=f(1)
f_s(0)=lim f(x)=lim(x*x+m*x+n)=n
       x->0    x->0
       x<0     x<0
f_d(0)=lim lim f(x)=lim(p*x*x+4*x+4)=4
       x->0    x->0
       x>0     x>0

f(0)=4
==>n=4==>
f(x)=x*x+m*x+4 for x in [-1,0)
f(x)=p*x*x+4*x+4 for x in [0,1]
f'_s(0)=lim f'(x)=lim(2*x+m)=m
       x->0    x->0
       x<0     x<0
f'_d(0)=lim lim f'(x)=lim(2*p*x+4)=4
       x->0    x->0
       x>0     x>0
==> m=4
==> f(x)=x*x+4*x+4 for x in [-1,0]
and f(x)=p*x*x+4*x+4 for x in [0,1]

But f(-1)=f(1)==>p=-7

So, n=4,m=4,p=-7

By Camelia Sima
Teacher: Ligia Garlea
"Alex. Papiu Ilarian" HS Dej, Cluj, Romania