23.4. EXERCISES 493

Note that in this simple example, the Hessian matrix is constant and so all that is leftis to consider the eigenvalues. Writing the characteristic equation and solving yields theeigenvalues are 2,−2,4. Thus the given point is a saddle point.

Remember that all you care about is the sign of the eigenvalues. You don’t have to findthem exactly.

23.4 Exercises1. Use the second derivative test on the critical points (1,1), and (1,−1) for Example

23.3.5. The function is 6xy2 −2x3 −3x4.

2. Show the points( 1

2 ,−214

),(0,−4) , and (1,−4) are critical points of the following

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

f (x,y) =−x4 +2x3 +39x2 +10yx2 −10yx−40x− y2 −8y−16.

3. Show the points( 1

2 ,−5312

),(0,−4) , and (1,−4) are critical points of the following

function of two variables and classify them according to whether they are local min-ima, local maxima or saddle points.

f (x,y) =−3x4 +6x3 +37x2 +10yx2 −10yx−40x−3y2 −24y−48.

4. Show the points( 1

2 ,3720

),(0,2) , and (1,2) are critical points of the following function

of two variables and classify them according to whether they are local minima, localmaxima or saddle points.

f (x,y) = 5x4 −10x3 +17x2 −6yx2 +6yx−12x+5y2 −20y+20.

5. Show the points( 1

2 ,−178

),(0,−2) , and (1,−2) are critical points of the following

function of two variables and classify them according to whether they are local min-ima, local maxima or saddle points.

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

6. 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) = 13 x2 + 32

3 x+ 43 −

163 yx− 58

3 y− 43 zx− 46

3 z+ 13 y2 − 4

3 zy− 53 z2.

7. 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) =− 53 x2 + 2

3 x− 23 +

83 yx+ 2

3 y+ 143 zx− 28

3 z− 53 y2 + 14

3 zy− 83 z2.

8. 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) =− 113 x2 + 40

3 x− 563 + 8

3 yx+ 103 y− 4

3 zx+ 223 z− 11

3 y2 − 43 zy− 5

3 z2.

9. 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.