Exercise · ZW-SECOND-DERIVATIVE-TEST-006 · ★★☆ Intermediate
For \(f(x,y)=x^2+xy+y^2\), compute \(D=f_{xx}f_{yy}-f_{xy}^{2}\) at the critical point \((0,0)\).
Choose one option.
Show answer
- D. 3
Explanation
\(f_{xx}=2\), \(f_{yy}=2\), \(f_{xy}=1\), so \(D=2\cdot2-1^2=3\). Forgetting to subtract \(f_{xy}^2\) gives 4.
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
- ZW-SECOND-DERIVATIVE-TEST-001 The function f(x,y)=x^2 + y^2 has a critical point at (0,0). What does the second derivative test s…
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- ZW-SECOND-DERIVATIVE-TEST-005 The function f(x,y)=x^4+y^4 has a critical point at (0,0). What does the second derivative test say?
- ZW-SECOND-DERIVATIVE-TEST-007 The function f(x,y)=x^2 + 3 x y + y^2 has a critical point at (0,0). What does the second derivativ…
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