Exercise · ZW-SECOND-DERIVATIVE-TEST-010 · ★★★ Advanced
At a critical point, \(D>0\) and \(f_{xx}>0\). What must be true about \(f_{yy}\)?
Choose one option.
Show answer
- C. \(f_{yy}>0\)
Explanation
\(D=f_{xx}f_{yy}-f_{xy}^2>0\) gives \(f_{xx}f_{yy}>f_{xy}^2\ge0\); since \(f_{xx}>0\), we need \(f_{yy}>0\).
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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