009A Sample Final A, Problem 10

10. Consider the function   $f(x)=2x^{3}+4x+{\sqrt {2}}.$
   (a) Use the Intermediate Value Theorem to show that $f(x)$ has at least one zero.
   (b) Use Rolle's Theorem to show that $f(x)$ has exactly one zero.

Foundations	The Intermediate Value Theorem. If f(x) is a continuous function on the interval [a,b], and if f(a) < f(b), then for any y such that
$f(a)\leq y\leq f(b),$
there exists a $c\in [a,b]$ such that $f(c)=y$.