The linear approximation to [tex]f(x,y,z)[/tex] about a point [tex](a,b,c)[/tex] is
[tex]L(x,y,z)=f(a,b,c)+\nabla f(a,b,c)\cdot\langle x-a,y-b,z-c\rangle[/tex]
We have
[tex]f(x,y,z)=x^2+y^2+z^2[/tex]
[tex]\implies\nabla f(x,y,z)=\langle2x,2y,2z\rangle[/tex]
so that near (6, 6, 7), we get [tex]f(x,y,z)[/tex] approximately equal to
[tex]L(x,y,z)=6^2+6^2+7^2+\langle12,12,14\rangle\cdot\langle x-6,y-6,z-7\rangle[/tex]
[tex]L(x,y,z)=12x+12y+14z-121[/tex]
Then
[tex]f(6.02,5.99,6.97)\approx L(6.02,5.99,6.97)=\boxed{120.70000}[/tex]
