Exercise · ZW-LOCAL-MAXIMA-AND-MINIMA-010 · ★★☆ Intermediate
At which value of x does \(f(x)=x^{4}-2x^{2}\) have a local maximum?
Choose one option.
Show answer
- D. \(0\)
Explanation
The critical points are \(x=0\) and \(x=\pm1\). With \(f''(x)=12x^{2}-4\), at \(x=0\) it equals \(-4<0\), so it is a local maximum; at \(x=\pm1\) it equals \(8>0\), so they are minima.
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-008 Is it true that if f'(c)=0, then c is a local extremum of f?