To minimize the function $f(x, y, z) = x^2 + y^2 + z^2$ subject to the constraint $g(x, y, z) = ax^2 + by^2 + cz^2 + 2fyz + 2qzx + 2hxy = 1$, we can apply the method of Lagrange multipliers.

The method states that the gradients of the function to be minimized ( $f$ ) and the constraint function ( $g$ ) must be proportional at the points of extremum. This gives us the following equation:

$\nabla f(x, y, z) = \lambda \nabla g(x, y, z)$

Step 1: Compute the gradients

• The gradient of $f(x, y, z) = x^2 + y^2 + z^2$ is: $\nabla f(x, y, z) = (2x, 2y, 2z)$

• The gradient of the constraint function $g(x, y, z) = ax^2 + by^2 + cz^2 + 2fyz + 2qzx + 2hxy$ is:

\frac{\partial g}{\partial x}, \frac{\partial g}{\partial y}, \frac{\partial g}{\partial z} \right)$$ The partial derivatives of $$g$$ are: - $$\frac{\partial g}{\partial x} = 2ax + 2qz + 2hy$$ - $$\frac{\partial g}{\partial y} = 2by + 2fz + 2hx$$ - $$\frac{\partial g}{\partial z} = 2cz + 2fy + 2qx$$ Thus, \[ \nabla g(x, y, z) = (2ax +