Find the minimum value of $f(x, y, z)=x^{2}+y^{2}+z^{2}$ subject to $-6 x-8 y+2 z=-52$
The value of $f$ at the minimum is (Simplify your answer.)

Step 1 ：The given function is \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) and the constraint is \(-6x-8y+2z=-52\).

Step 2 ：We can rewrite the constraint as \(z=3x+4y+26\).

Step 3 ：Substitute \(z\) into the function, we get \(f(x, y, z)=x^{2}+y^{2}+(3x+4y+26)^{2}\).

Step 4 ：Expand the function, we get \(f(x, y, z)=x^{2}+y^{2}+9x^{2}+16y^{2}+156x+208y+676\).

Step 5 ：Combine like terms, we get \(f(x, y, z)=10x^{2}+17y^{2}+156x+208y+676\).

Step 6 ：To find the minimum value of the function, we need to find the derivative of the function and set it equal to zero.

Step 7 ：The derivative of the function with respect to \(x\) is \(20x+156\), and the derivative of the function with respect to \(y\) is \(34y+208\).

Step 8 ：Setting these derivatives equal to zero, we get \(x=-7.8\) and \(y=-6.11764705882\).

Step 9 ：Substitute \(x\) and \(y\) into the function, we get \(f(-7.8, -6.11764705882, z)=10*(-7.8)^{2}+17*(-6.11764705882)^{2}+156*(-7.8)+208*(-6.11764705882)+676\).

Step 10 ：Calculate the value, we get \(f(-7.8, -6.11764705882, z)=676\).

Step 11 ：Therefore, the minimum value of the function is \(\boxed{676}\).