Smallest value

Ans) 10
Level= jee advance


apply the Minkowski inequality, now check when the equal sign hold.
$x_{2}+1 +y_{2}+4 +z_{2}+9 $ >
$(x+y+z)_{2}+(1+2+3)_{2} $not this is not in jee syllabus

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how does this solution gurantees existence of minimum value of expresssion?

@ashishdeosingh since √(x²+1) + √(y²+4) + √(z²+9) are on same line which is the only possible way to get minimum value...
And also it forms right angled triangle in which hypotenuse is the minimum value which we want

not a standard method.
its like solution made first then question made.