009A Sample Final A, Problem 10
10. Consider the function
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=2x^{3}+4x+\sqrt{2}.}
(a) Use the Intermediate Value Theorem to show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)}
has at
least one zero.
(b) Use Rolle's Theorem to show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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. |