Exercise · ZW-INTERMEDIATE-VALUE-THEOREM-001 · ★☆☆ Basic
Let \(f(x)=x^{3}+x-1\). On which interval does the intermediate value theorem guarantee a root?
Choose one option.
Show answer
- A. \([0,1]\)
Explanation
f is continuous on all of ℝ. Only on \([0,1]\) do the endpoint values have opposite signs (f(0) = -1, f(1) = 1), so by the intermediate value theorem there is a root there. On the other intervals the signs agree and the theorem guarantees nothing.
Review the concept: Intermediate value theorem
If \(f\) is continuous on \([a,b]\) and \(N\) lies between \(f(a)\) and \(f(b)\), there is some \(c\in(a,b)\) with \(f(c)=N\).
\[f\ \text{continuous on }[a,b],\ f(a)
More practice problems: intermediate value theorem
- ZW-INTERMEDIATE-VALUE-THEOREM-002 Let f(x)=x^2-3. On which interval does the intermediate value theorem guarantee a root?
- ZW-INTERMEDIATE-VALUE-THEOREM-003 Let f(x)=cos x-x. On which interval does the intermediate value theorem guarantee a root?
- ZW-INTERMEDIATE-VALUE-THEOREM-004 Which hypothesis on f does the intermediate value theorem require on [a,b]?
- ZW-INTERMEDIATE-VALUE-THEOREM-005 If N lies between f(a) and f(b), what does the intermediate value theorem state?
- ZW-INTERMEDIATE-VALUE-THEOREM-006 Let f(x)=x^3-x-1. On which interval does the intermediate value theorem guarantee a root?
- ZW-INTERMEDIATE-VALUE-THEOREM-007 Let f be continuous on [0,4] with f(0)=2 and f(4)=-1. What can be guaranteed?
- ZW-INTERMEDIATE-VALUE-THEOREM-008 Let f(x)=1/x on [-1,1]. Can the theorem be used to find c with f(c)=0?
- ZW-INTERMEDIATE-VALUE-THEOREM-009 What kind of result is the intermediate value theorem?