Exercise · ZW-INTERMEDIATE-VALUE-THEOREM-013 · ★☆☆ Basic
Let \(f(x)=\cos x\) on \([0,\pi]\). For which value \(y\) does the theorem guarantee some \(c\) with \(f(c)=y\)?
Choose one option.
Show answer
- A. \(y=-\tfrac12\)
Explanation
Since \(f(0)=1\) and \(f(\pi)=-1\), only values in \([-1,1]\) are guaranteed; among the options, only \(-\tfrac12\) is there.
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?
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