494 CHAPTER 23. OPTIMIZATION

10. ∗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.

11. 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.

12. 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.

13. 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.

14. 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.

15. 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 − 10

3 yx+ 163 y+ 2

3 zx− 203 z+ 10

3 y2 + 23 zy+ 4

3 z2.

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) =− 73 x2 − 146

3 x+ 833 + 16

3 yx+ 43 y− 14

3 zx+ 943 z− 7

3 y2 − 143 zy+ 8

3 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) = 23 x2 +4x+75− 14

3 yx−38y− 83 zx−2z+ 2

3 y2 − 83 zy− 1

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

19. Show 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.

20. Show that the critical points of the following function are

(0,−3,0) ,(2,−3,0) ,and(

1,−3,−13

)