Engineering Calculus Notes 401

Engineering Calculus Notes 401 - no information. As an...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
3.9. THE PRINCIPAL AXIS THEOREM 389 Theorem 3.9.8 (Second Derivative Test, Three Variables) . Suppose the C 2 function f ( −→ x ) = f ( x,y,z ) has a critical point at −→ x = −→ a . Consider the following three quantities: Δ 1 = f xx ( −→ a ) Δ 2 = f xx ( −→ a ) f yy ( −→ a ) f xy ( −→ a ) 2 Δ 3 = det Hf ( −→ a ) . 1. If Δ k > 0 for k = 1 , 2 , 3 , then f ( −→ x ) has a local minimum at −→ x = −→ a . 2. If ( 1) k Δ k > 0 for k = 1 , 2 , 3 (i.e., , Δ 2 > 0 while Δ 1 < 0 and Δ 3 < 0 ), then f ( −→ x ) has a local maximum at −→ x = −→ a . 3. If all three quantities are nonzero but neither of the preceding conditions holds, then f ( −→ x ) does not have a local extremum at −→ x = −→ a . A word of warning: when one of these quantities equals zero, this test gives
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: no information. As an example of the use of Theorem 3.9.8 , consider the function f ( x,y,z ) = 2 x 2 + 2 y 2 + 2 z 2 + 2 xy + 2 xz + 2 yz 6 x + 2 y + 4 z ; its partial derivatives are f x ( x,y,z ) = 4 x + 2 y + 2 z 6 f y ( x,y,z ) = 4 y + 2 x + 2 z + 2 f z ( x,y,z ) = 4 z + 2 y + 2 x + 4 . Setting all of these equal to zero, we have the system of equations 4 x +2 y +2 z = 6 2 x +4 y +2 z = 2 2 x +2 y +4 z = 4 whose only solution is x = 3 y = 1 z = 2 ....
View Full Document

Ask a homework question - tutors are online