Exercise · ZW-SECOND-DERIVATIVE-TEST-008 · ★★☆ Intermediate
For \(f(x,y)=x^3+y^3-3xy\), the point \((1,1)\) is critical. What does the test say there?
Choose one option.
Show answer
- A. Local minimum
Explanation
At \((1,1)\): \(f_{xx}=6\), \(f_{yy}=6\), \(f_{xy}=-3\); \(D=36-9=27>0\) and \(f_{xx}>0\): local minimum. (At \((0,0)\), \(D=-9\): saddle.)
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-006 For f(x,y)=x^2+xy+y^2, compute D=f_xxf_yy-f_xy^2 at the critical point (0,0).
- 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…
- ZW-SECOND-DERIVATIVE-TEST-009 At a critical point of f we find D>0 and f_xx<0. What kind of point is it?
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