[Math] Show that function $4x^5+x^3+2x+1=0$ has exactly one real root.

Question

I am struggling with problems like this, just trying to grasp the concept.
Using the Intermediate Value Theorem and Rolle's Theorem, prove that $4x^5+x^3+2x+1=0$ has exactly one real root.

Answer 1

$f'(x) = 20x^4 + 3x^2 + 2 > 0$ for all real values $x$, and $f(0) = 1 > 0$, $f(-1) = -6 < 0$, then by the intermediate value theorem there exits one and only one real root in $(-1,0)$ for the equation: $4x^5 + x^3 + 2x + 1 = 0$