Exercise · ZW-LOCAL-MAXIMA-AND-MINIMA-008 · ★★★ Advanced
Is it true that if \(f'(c)=0\), then \(c\) is a local extremum of \(f\)?
Choose one option.
Show answer
- B. No: for \(f(x)=x^{3}\) at \(c=0\), \(f'(0)=0\) but there is no extremum
Explanation
It is false: \(f'(x)=3x^{2}\geq 0\) does not change sign at \(0\), so \(f\) is increasing and \(x=0\) is not an extremum. Setting \(f'=0\) is necessary but not sufficient.
Review the concept: Local maxima and minima
A local maximum of \(f\) at \(c\) is a value \(f(c)\) greater than or equal to the values of \(f\) on some neighborhood of \(c\); a local minimum, less than or equal.
\[f'(c)=0,\ f''(c)>0\ \Rightarrow\ \text{local minimum}\]
More practice problems: local maxima and minima
- ZW-LOCAL-MAXIMA-AND-MINIMA-001 At which value of x does f(x)=x^3 - 3 x have a local maximum?
- ZW-LOCAL-MAXIMA-AND-MINIMA-002 At which value of x does f(x)=2 x^3 - 9 x^2 + 12 x have a local maximum?
- ZW-LOCAL-MAXIMA-AND-MINIMA-003 At which value of x does f(x)=x^3 - 12 x + 5 have a local maximum?
- ZW-LOCAL-MAXIMA-AND-MINIMA-004 At which value of x does f(x)=- x^2 + 6 x have a local maximum?
- ZW-LOCAL-MAXIMA-AND-MINIMA-005 When is c a critical point of f?
- ZW-LOCAL-MAXIMA-AND-MINIMA-006 What does it mean for f to have a local maximum at c?
- ZW-LOCAL-MAXIMA-AND-MINIMA-007 What kind of extremum does f(x)=x^2-4x+1 have at x=2?
- ZW-LOCAL-MAXIMA-AND-MINIMA-009 Which test classifies a critical point by the sign of f''?
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