# If the last condition is not satisfied or if and have

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If the last condition is not satisfied , or if and have opposite signs , the stationary point is a saddle point . Examples: Solve the first and second order conditions for the following functions - which we partially differentiated in Section 29 - and identify any stationary points: (i) y = x + 2 z + 4 xz 2 2 (ii) f ( x, z ) = 4 xz - x - 2 z 2 4 1 2 1 1 2 2 1 2 (iii) f ( x , x ) = x + 4 x x + 2 x - 2 x - x 2 2

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83 31. A Mathematical Model of Mountains ! Recall Green Gable, Great Gable: and Windy Gap... ... and the Ordnance Survey map with contour lines showing height above sea level:
84 ! The function below when graphed in three dimensions looks a bit like Green Gable, Great Gable and Windy Gap, and will be used in Exercise 39. y = 1250 - 50(x-5) + 1080z - 438z + 64z - 3z 2 2 3 4 for 0 < x < 10 and 0 < z < 11 ! and the same graphic with contour lines: Note: Ignore the vertical scales in the above diagrams, which have been exaggerated in order to make them clearer.

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85 TUTORIAL EXERCISES 37. For the three functions below, whose three-dimensional graphs featured in Section 27, show that: (i) f ( x, z ) = 8 - ( x - 2) - ( z - 2) has a maximum, 2 2 (iii) f ( x, z ) = 8 + ( x - 2) + ( z - 2) has a minimum, and 2 2 (iv) f ( x, z ) = 8 + ( x - 2) - ( z - 2) has a saddle point. 2 2 38. Show that the following function below (which you will notice is similar to the function in 37(i) ) has a saddle point, not a maximum, where x = 2 and z = 2. f ( x, z ) = 8 - ( x - 2) - ( z - 2) + 3( x - 2)( z - 2) 2 2 39. In section 31, the function below was used to model the mountains Green Gable and Great Gable in the Lake District and the saddle Windy Gap between them: y = 1250 - 50(x-5) + 1080z - 438z + 64z - 3z 2 2 3 4 Solve the first and second order conditions for the function and show that the ( x,z ) coordinates of “Green Gable”, “Windy Gap” and “Great Gable” are, respectively, (5, 2), (5, 5) and (5, 9).
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