EXAMPLES OF MAXIMA AND MINIMA. 245 



x and y which make -y- and -7- vanish are x = Q. w = 0: and 

 dx ay 



these give a minimum value to u. 



7. Find the maximum value of x*y* (<oxy). 



Result. Maximum when x = 3, y = 2. 



8. If u = (2aa; y?} (2by y*}, find its maximum or mini- 



mum value. 



Result. x = a, y = b, make u a maximum. 



9. If u = x* + y* 2x* + kxy 2^ 2 , shew that when x = 0, and 



y = 0, u is neither a maximum nor minimum ; when 

 x = v% and y= + */2, u is a minimum. 



10. If w =# 4 - 8# 3 + ISy 2 - 8y + x 3 - 3x 2 - Bx, then 3 + 4 V2 



is a maximum value of u and 6 4 A/2 is a minimum 

 value of w. 



11. If u = a? + xy +y z ax by, then ^(ab a? 5 2 ) is a 



minimum value of u. 



12. Divide a number n into three parts, #, y, and 2, such 



,, , a?y #3 #3 T n i 

 that + + ^T- shall be 



O 4 



and determine which it is. 



,, , a?y #3 #3 T n i 



that + + ^T- shall be a maximum or mmimum, 



O 4 



/y fti flj 



Result. = -~^ = - a maximum. 



13. If u = x z + y s + Saxy, then a 3 is a maximum value of u. 



14. Find the maximum or minimum of x (x* + y z ] Boxy. 



15. Find the maximum or minimum of j-. 



Result. - = V = r: 2 , 72 > 



a 6 a + 6 



with the upper sign there is a maximum, with the 

 lower a minimum. 



16. If u = *J{(c-x}(c-y)(x + y-c)}, shew that it is a 



2c 



maximum when x=y= . 

 o 



