Exercise · ZW-SECOND-DERIVATIVE-TEST-003 · ★☆☆ Basic
The function \(f(x,y)=- x^{2} + 4 x - y^{2}\) has a critical point at \((2,0)\). What does the second derivative test say?
Choose one option.
Show answer
- B. Local maximum
Explanation
\(f_{xx}=-2\), \(f_{yy}=-2\), \(f_{xy}=0\); \(D=4>0\) and \(f_{xx}<0\), so it 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).
- 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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