Exercise · ZW-INTERMEDIATE-VALUE-THEOREM-011 · ★☆☆ Basic

Let \(f(x)=x^3-x-1\) on \([1,2]\). Since \(f(1)=-1\) and \(f(2)=5\), what does the intermediate value theorem guarantee?

Choose one option.

Show answer

B. That there is \(c\in[1,2]\) with \(f(c)=0\)

Explanation

Zero lies between \(-1\) and \(5\), so some \(c\) attains it; the theorem does not say where it is or that \(f\) is monotone, and \(7>5\) is out of range.

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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-001 Let f(x)=x^3+x-1. On which interval does the intermediate value theorem guarantee a root?
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?