Prove that f(x)=x^5+2x^3+4x-12 has exactly one real root.

Question

Hi, Im having trouble proving this. I only got half way through proving it. Its the Rolles Theorem part Im having trouble with. So first thing I did was show that there is a root by the IVT. f(0)= -12 which is less than zero. And f(2)= 44 which is greater than zero so by the IVT there exists a number in the interval ( 0,2) such that f(c)=0. 

 

The next part is showing with Rolles theorem but this where Im lost. 

Answer 1

By College Algebra (Descartes' rule of signs), there is exactly one Real positive root.

 

Since f(-x) =- x^5 - 2x^3 - 4x-12 and there is no sign change, there are no Real negative roots.

 

QED (w/o Calculus).