Exercise · ZW-LOCAL-MAXIMA-AND-MINIMA-006 · ★☆☆ Basic
What does it mean for \(f\) to have a local maximum at \(c\)?
Choose one option.
Show answer
- D. \(f(c)\geq f(x)\) for all \(x\) near \(c\)
Explanation
There is a local maximum at \(c\) if \(f(c)\) is at least as large as the values of \(f\) on some neighborhood of \(c\). Only nearby points matter, not the whole domain.
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-007 What kind of extremum does f(x)=x^2-4x+1 have at x=2?
- ZW-LOCAL-MAXIMA-AND-MINIMA-008 Is it true that if f'(c)=0, then c is a local extremum of f?
- ZW-LOCAL-MAXIMA-AND-MINIMA-009 Which test classifies a critical point by the sign of f''?
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