6.10. EXERCISES 179

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) = − 32x

4 + 6x3 − 6x2 + zx2 − 2zx− 2y2 − 12y − 18− 32z

2.

21. Here is a function of three variables.

f (x, y, z) = 13x2 + 2xy + 8xz + 13y2 + 8yz + 10z2

change the variables so that in the new variables there are no mixed terms, termsinvolving xy, yz etc. Two eigenvalues are 12 and 18.

22. Here is a function of three variables.

f (x, y, z) = 2x2 − 4x+ 2 + 9yx− 9y − 3zx+ 3z + 5y2 − 9zy − 7z2

change the variables so that in the new variables there are no mixed terms, termsinvolving xy, yz etc. The eigenvalues of the matrix which you will work with are− 17

2 ,192 ,−1.

23. Here is a function of three variables.

f (x, y, z) = −x2 + 2xy + 2xz − y2 + 2yz − z2 + x

change the variables so that in the new variables there are no mixed terms, termsinvolving xy, yz etc.

24. Show the critical points of the function,

f (x, y, z) = −2yx2 − 6yx− 4zx2 − 12zx+ y2 + 2yz.

are points of the form,

(x, y, z) =(t, 2t2 + 6t,−t2 − 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

2x4 − 4x3 + 8x2 − 3zx2 + 12zx+ 2y2 + 4y + 2 +

1

2z2.

are (0,−1, 0) , (4,−1, 0) , and (2,−1,−12) and classify them as local minima, localmaxima or saddle points.