Exercise · ZW-SECOND-DERIVATIVE-TEST-009 · ★★★ Advanced
At a critical point of \(f\) we find \(D>0\) and \(f_{xx}<0\). What kind of point is it?
Choose one option.
Show answer
- B. Local maximum
Explanation
\(D>0\) rules out a saddle and the sign of \(f_{xx}\) decides: negative means downward curvature, that is, a local maximum.
Review the concept: Second derivative test
The second derivative test classifies a critical point of \(f\) as a minimum, maximum or saddle according to the sign of the discriminant \(D\) and of \(f_{xx}\) at that point.
\[D=f_{xx}f_{yy}-\left(f_{xy}\right)^{2}\]
More practice problems: second derivative test
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- ZW-SECOND-DERIVATIVE-TEST-006 For f(x,y)=x^2+xy+y^2, compute D=f_xxf_yy-f_xy^2 at the critical point (0,0).
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