1. ## Finding Real Roots

Hello, and thanks for the help in advance!

I have several problems that are all in the same vein as the one below, but I don't have much of an idea as to how to solve them. I would apply the quadratic formula if exponents allowed, because that is how I am accustomed to finding roots, but these problems don't seem to allow for that.

Show that the following equation has exactly one real root:

1 + 2x + x^3 + 4x^5 = 0

Thanks again for the help...my only other idea is to take the derivative of both sides, but I'm not sure where this would lead us...

2. Hint: Letting $f(x)=4x^5+x^3+2x+1,$ show that $f'(x)>0$ for all $x\in\mathbb R.$ What does this tell you about $f(x)\,?$