Question

The minimum value, \(f_{\text{min}}\), of the function given below is: \[ f(x_1, x_2, x_3) = \frac{1}{2} (x_1^2 + x_2^2 + x_3^2) - 2(x_1 + x_2 + x_3) \] (round off to the nearest integer).

Hint:

To minimize a multivariable function, set partial derivatives equal to zero and solve for the variables.

Answer 1

Correct Answer: -6

To minimize the function, we take partial derivatives with respect to \(x_1\), \(x_2\), and \(x_3\) and set them to zero: \[ \frac{\partial f}{\partial x_1} = x_1 - 2 = 0 $\Rightarrow$ x_1 = 2 \] \[ \frac{\partial f}{\partial x_2} = x_2 - 2 = 0 $\Rightarrow$ x_2 = 2 \] \[ \frac{\partial f}{\partial x_3} = x_3 - 2 = 0 $\Rightarrow$ x_3 = 2 \] Substitute \(x_1 = x_2 = x_3 = 2\) into the original function: \[ f_{\text{min}} = \frac{1}{2} (2^2 + 2^2 + 2^2) - 2(2 + 2 + 2) \] \[ = \frac{1}{2}(12) - 12 = 6 - 12 = -6 \] Thus, \[ \boxed{-6} \]