System of equations, I think...

[palangi]

I can not remember how I am supposed begin this. Will someone please help? Could you also include the final answer so I can make sure I did it correctly? Thank you.

If a and b are positive real numbers prove that the equation a/(x^3*2x^2-1) + b/(x^3+x-2) has at least one solution in the interval (-1,1)

 

[galactus]

I can not remember how I am supposed begin this. Will someone please help? Could you also include the final answer so I can make sure I did it correctly? Thank you.

If a and b are positive real numbers prove that the equation a/(x^3*2x^2-1) + b/(x^3+x-2) has at least one solution in the interval (-1,1)

Is that supposed to be \(\displaystyle \frac{a}{x^{3}+2x^{2}-1}+\frac{b}{x^{3}+x-2}\) ?.

I think you have a * where a + should be?.

 

[palangi]

yes, that's it

 

[mmm4444bot]

Each of the given denominators is the product of a linear and quadratic factor.

The roots of these factors show that the given expression is a continuous Rational function within the interval (-1,1) -- except at the vertical asymptote x = [sqrt(5) - 1]/2 -- because ratios of polynomials are continuous wherever the denominator is not zero.

My first approach would be to combine the given ratios into a single ratio, followed by analyzing sign changes in the resulting numerator on one side of the asymptote (then the other, if needed), as I suspect that the sign of the resulting denominator does not change on the same side.

In other words, showing that the numerator changes sign, while the denominator does not, within either interval (-1,Z) or (Z,1), proves that the graph must cross the x-axis somewhere.

I used Z = [sqrt(5) - 1]/2 in my interval notation above.