Given that x + y + z = 10 we need to find the maximum value of 1/x + 4/y + 9/z.
We have to assume that x, y and z are distinct and have integral values to be able to derive the maximum value of 1/x + 4/y + 9/z. If they can have any value the expression can have a value that tends to infinity.
Working under the constraints mentioned, we take z as the smallest integer between 0 and 10. That makes z = 1, y is the integer that follows 1 or 2 and x = 10 - 3 = 7.
This gives 1/x + 4/y + 9/z = 1/7 + 4/2 + 9/1 = 1/7 + 2 + 9 = 78/7
If x, y and z have integral values the maximum value of 1/x + 4/y + 9/z = 78/7
We choose x=9.9, y = 1/0.05 and z = 1/0.05, then 1/x+4/y+9/0.05 = 1/10+4/0.05+9/0.05 = 260
We can choose x as near 10 and make y and z positive and smaller. Then 1/x+4/y+9/z become larger and unbounded
We make (10-x) -> - 0, y-> +0, z -> +0, then obviously, 1/x+4/y+9/z approaches infinity
Therefore the expression 1/x+y/4+9/z has no maximum.