f = x+y+z+k( 1/x+ 1/y+ 1/z- 1)
F ' & ltx & gt= 0: 1-k/x 2 = 0，x2 = k；
F ' & lty & gt= 0: 1-k/y ^ 2 = 0，y ^ 2 = k；
F ' & ltz & gt= 0: 1-k/z 2 = 0，z2 = k；
F ' & ltk & gt= 0: 1/x+ 1/y+ 1/z = 1
Then 3/√k = 1, or1√ k =1,
The solution is k= 9 or k= 1.
Maximum x+y+z = 27.
The minimum value x+y+z = -27.
The good habits of learning the important knowledge of junior high school mathematics should be: asking more questions, thinking hard