In this problem you will use Rolle's theorem to determine whether it is possible for the function $$f(x) = -\left(6x^{5}+8x+19\right)$$ to have two or more real roots (or, equivalently, whether the graph of $y = f(x)$ crosses the $x$-axis two or more times).

Suppose that $f(x)$ has at least two real roots. Choose two of these roots and call the smaller one $a$ and the larger one $b$. By applying Rolle's theorem to $f(x)$ on the interval $[a,b]$, there exists at least one number $c$ in the interval $(a,b)$ so that $f'(c) =$ .

The values of the derivative $f'(x) =$ are always ? changing negative zero positive undefined , and therefore it is ? plausible unlikely possible impossible for $f(x)$ to have two or more real roots.