Exercise · ZW-INTERMEDIATE-VALUE-THEOREM-015 · ★★☆ Intermediate

Why can the theorem not be used to ensure a zero of \(f(x)=1/x\) on \([-1,1]\), even though \(f(-1)=-1\) and \(f(1)=1\)?

Choose one option.

Show answer

A. Because \(f\) is neither defined nor continuous at \(0\)

Explanation

The theorem requires continuity on the whole interval; \(1/x\) does not exist at \(0\) and in fact never equals \(0\). The signs at the endpoints are indeed opposite.

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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?