336 CHAPTER 18. OPTIMIZATION

10. Find the critical points of the following function of three variables and classify themaccording to whether they are local minima, local maxima or saddle points.

f (x,y,z) =− 23 x2 + 28

3 x+ 373 + 14

3 yx+ 103 y− 4

3 zx− 263 z− 2

3 y2− 43 zy+ 7

3 z2.

11. ∗Show that if f has a critical point and some eigenvalue of the Hessian matrix ispositive, then there exists a direction in which when f is evaluated on the line throughthe critical point having this direction, the resulting function of one variable has alocal minimum. State and prove a similar result in the case where some eigenvalueof the Hessian matrix is negative.

12. Suppose µ = 0 but there are negative eigenvalues of the Hessian at a critical point.Show by giving examples that the second derivative tests fails.

13. Show that the points( 1

2 ,−92

),(0,−5), and (1,−5) are critical points of the following

function of two variables and classify them as local minima, local maxima or saddlepoints.

f (x,y) = 2x4−4x3 +42x2 +8yx2−8yx−40x+2y2 +20y+50.

14. Show that the points(1,− 11

2

),(0,−5), and (2,−5) are critical points of the follow-

ing function of two variables and classify them as local minima, local maxima orsaddle points.

f (x,y) = 4x4−16x3−4x2−4yx2 +8yx+40x+4y2 +40y+100.

15. Show that the points( 3

2 ,2720

),(0,0), and (3,0) are critical points of the following

function of two variables and classify them as local minima, local maxima or saddlepoints.

f (x,y) = 5x4−30x3 +45x2 +6yx2−18yx+5y2.

16. Find the critical points of the following function of three variables and classify themas local minima, local maxima or saddle points.

f (x,y,z) = 103 x2− 44

3 x+ 643 −

103 yx+ 16

3 y+ 23 zx− 20

3 z+ 103 y2 + 2

3 zy+ 43 z2.

17. Find the critical points of the following function of three variables and classify themas local minima, local maxima or saddle points.

f (x,y,z) =− 73 x2− 146

3 x+ 833 + 16

3 yx+ 43 y− 14

3 zx+ 943 z− 7

3 y2− 143 zy+ 8

3 z2.

18. Find the critical points of the following function of three variables and classify themas local minima, local maxima or saddle points.

f (x,y,z) = 23 x2 +4x+75− 14

3 yx−38y− 83 zx−2z+ 2

3 y2− 83 zy− 1

3 z2.

19. Find the critical points of the following function of three variables and classify themas local minima, local maxima or saddle points.

f (x,y,z) = 4x2−30x+510−2yx+60y−2zx−70z+4y2−2zy+4z2.

20. Show that the critical points of the following function are points of the form, (x,y,z)=(t,2t2−10t,−t2 +5t

)for t ∈ R and classify them as local minima, local maxima or

saddle points.

f (x,y,z) =− 16 x4 + 5

3 x3− 256 x2 + 10

3 yx2− 503 yx+ 19

3 zx2− 953 zx− 5

3 y2− 103 zy− 1

6 z2.