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) ≤ y ≤ f(b), then there exists a c ∈ [a,b] such that f(c) = y. Similarly, if f(a) ≥ f(b), then for any y such that f(a) ≥ y ≥ f(b), then there exists a c ∈ [a,b] such that f(c) = y.
In part (a) of this problem, as many others, we are trying to show that a root or zero exists. In order to apply the IVT, we need to note that the function is continuous, and then find an a and b such that, for example, f(a) < 0, while f(b) > 0.