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
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