1. **Problem Statement:** Find the minimum value of $x^2 + y^2 + z^2$ subject to the constraint $xyz = a^3$ using Lagrange multipliers. 2. **Formula and Concept:** To find extrema of a function $f(x,y,z)$ subject to a constraint $g(x,y,z) = 0$, use Lagrange multipliers: $$\nabla f = \lambda \nabla g$$ where $\lambda$ is the Lagrange multiplier. 3. **Set up the problem:** - Objective function: $f(x,y,z) = x^2 + y^2 + z^2$ - Constraint: $g(x,y,z) = xyz - a^3 = 0$ 4. **Calculate gradients:** $$\nabla f = (2x, 2y, 2z)$$ $$\nabla g = (yz, xz, xy)$$ 5. **Apply Lagrange multiplier condition:** $$2x = \lambda yz$$ $$2y = \lambda xz$$ $$2z = \lambda xy$$ 6. **Solve for $\lambda$ and variables:** From the first two equations: $$\frac{2x}{yz} = \frac{2y}{xz} \implies \frac{x}{yz} = \frac{y}{xz} \implies x^2 = y^2$$ Similarly, comparing other pairs gives: $$x^2 = y^2 = z^2$$ 7. **Choose positive values (since squares equal):** $$x = y = z$$ 8. **Use constraint to find $x$:** $$xyz = x^3 = a^3 \implies x = a$$ 9. **Find minimum value:** $$f = x^2 + y^2 + z^2 = 3a^2$$ **Final answer:** The minimum value of $x^2 + y^2 + z^2$ subject to $xyz = a^3$ is $$\boxed{3a^2}$$.