1. **State the problem:** Find the extrema values of the function $$f(x,y,z) = x^2 + 2y^2 + 3z^2$$ subject to the constraints $$x + y + z = 1$$ and $$x - y = 0$$. 2. **Use the method of Lagrange multipliers:** For constraints $$g_1(x,y,z) = x + y + z - 1 = 0$$ and $$g_2(x,y,z) = x - y = 0$$, set up the system: $$\nabla f = \lambda \nabla g_1 + \mu \nabla g_2$$ 3. **Calculate gradients:** $$\nabla f = (2x, 4y, 6z)$$ $$\nabla g_1 = (1, 1, 1)$$ $$\nabla g_2 = (1, -1, 0)$$ 4. **Set up equations:** $$2x = \lambda + \mu$$ $$4y = \lambda - \mu$$ $$6z = \lambda$$ 5. **Use constraints:** From $$x - y = 0$$, we get $$x = y$$. From $$x + y + z = 1$$, substitute $$y = x$$: $$2x + z = 1$$ 6. **Express $$\lambda$$ and $$\mu$$ in terms of $$x$$ and $$z$$:** From the first two equations: $$2x = \lambda + \mu$$ $$4x = \lambda - \mu$$ Add: $$2x + 4x = 2\lambda \Rightarrow 6x = 2\lambda \Rightarrow \lambda = 3x$$ Subtract: $$2x - 4x = 2\mu \Rightarrow -2x = 2\mu \Rightarrow \mu = -x$$ 7. **Use third equation:** $$6z = \lambda = 3x \Rightarrow 6z = 3x \Rightarrow 2z = x$$ 8. **Use constraint $$2x + z = 1$$:** Substitute $$z = \frac{x}{2}$$: $$2x + \frac{x}{2} = 1 \Rightarrow \frac{4x + x}{2} = 1 \Rightarrow \frac{5x}{2} = 1 \Rightarrow x = \frac{2}{5}$$ 9. **Find $$y$$ and $$z$$:** $$y = x = \frac{2}{5}$$ $$z = \frac{x}{2} = \frac{1}{5}$$ 10. **Calculate $$f(x,y,z)$$:** $$f\left(\frac{2}{5}, \frac{2}{5}, \frac{1}{5}\right) = \left(\frac{2}{5}\right)^2 + 2\left(\frac{2}{5}\right)^2 + 3\left(\frac{1}{5}\right)^2 = \frac{4}{25} + 2 \cdot \frac{4}{25} + 3 \cdot \frac{1}{25} = \frac{4}{25} + \frac{8}{25} + \frac{3}{25} = \frac{15}{25} = \frac{3}{5}$$ **Final answer:** The extrema value of $$f$$ subject to the constraints is $$\frac{3}{5}$$ at $$\left(\frac{2}{5}, \frac{2}{5}, \frac{1}{5}\right)$$.