Linear Algebra, Theory And Applications, 2012a

192 SPECTRAL THEORY
19. Here is a matrix. ⎛
⎜
⎝
1234 6 5 3
0 −654 9 123
98 123 10, 000 11
56 78 98 400
I know this matrix has an inverse before doing any computations. How do I know?
20. Show the critical points of the following function are
(
(0, −3, 0) , (2, −3, 0) , and 1, −3, − 1 )
3
and classify them as local minima, local maxima or saddle points.
f (x, y, z) =− 3 2 x4 +6x 3 − 6x 2 + zx 2 − 2zx − 2y 2 − 12y − 18 − 3 2 z2 .
21. Here is a function of three variables.
⎞
⎟
⎠
f (x, y, z) =13x 2 +2xy +8xz +13y 2 +8yz +10z 2
change the variables so that in the new variables there are no mixed terms, terms
involving xy, yz etc. Two eigenvalues are 12 and 18.
22. Here is a function of three variables.
f (x, y, z) =2x 2 − 4x +2+9yx − 9y − 3zx +3z +5y 2 − 9zy − 7z 2
change the variables so that in the new variables there are no mixed terms, terms
involving xy, yz etc. The eigenvalues of the matrix which you will work with are
− 17 2 , 19 2 , −1.
23. Here is a function of three variables.
f (x, y, z) =−x 2 +2xy +2xz − y 2 +2yz − z 2 + x
change the variables so that in the new variables there are no mixed terms, terms
involving xy, yz etc.
24. Show the critical points of the function,
f (x, y, z) =−2yx 2 − 6yx − 4zx 2 − 12zx + y 2 +2yz.
are points of the form,
(x, y, z) = ( t, 2t 2 +6t, −t 2 − 3t )
for t ∈ R and classify them as local minima, local maxima or saddle points.
25. Show the critical points of the function
f (x, y, z) = 1 2 x4 − 4x 3 +8x 2 − 3zx 2 +12zx +2y 2 +4y +2+ 1 2 z2 .
are (0, −1, 0) , (4, −1, 0) , and (2, −1, −12) and classify them as local minima, local
maxima or saddle points.
26. Let f (x, y) =3x 4 − 24x 2 +48− yx 2 +4y. Find and classify the critical points using
the second derivative test.