if a and b are positive, prove:

a/(x^3 - 2x^2 - 1) + b/(x^3 + x - 2) = 0

has at least one solution in the interval (-1,1)

Question

if a and b are positive, prove:

a/(x^3 - 2x^2 - 1) + b/(x^3 + x - 2) = 0

has at least one solution in the interval (-1,1)

anonymous · Answer

please help??

debbieg · Answer

Here is what I'm thinking.... over a common den'r the numerator would be:

a(x^3+x-2)+b(x^3-2x^2-1)

Which is a 3rd degree polynomial, so continuous for all x.

Now evaluate for x=1 and for x=-1. I suspect that one will be positive and the other negative. That proves that the numerator has a 0 in that interval.

debbieg · Answer

So for the numerator you get:$$\large a(x^3+x-2)+b(x^3-2x^2-1)=ax^3+ax-2a+bx^3-2bx^2-b$$$$\large =(a+b)x^3-2bx^2+ax-2a-b$$
Call this N(x). Then:
N(1)=(a+b)-2b+a-2a-b=-2b<0
N(-1)=-(a+b)-2b-a-2a-b=-4a-4b=-4(a+b)<0

Oooook..... I was so sure that would work. Never mind. Hopefully someone else has some ideas.