1. **State the problem:** Minimize the function $$f(x,y) = (x-1)^2 + y^2$$ subject to the constraint $$y - 2x = 0$$. 2. **Use the constraint to express one variable in terms of the other:** From $$y - 2x = 0$$, we get $$y = 2x$$. 3. **Substitute the constraint into the function:** $$f(x) = (x-1)^2 + (2x)^2 = (x-1)^2 + 4x^2$$ 4. **Simplify the function:** $$f(x) = (x^2 - 2x + 1) + 4x^2 = 5x^2 - 2x + 1$$ 5. **Find the critical points by differentiating and setting to zero:** $$f'(x) = 10x - 2$$ Set $$f'(x) = 0$$: $$10x - 2 = 0$$ $$10x = 2$$ $$x = \frac{2}{10} = \frac{1}{5}$$ 6. **Find corresponding $$y$$ using the constraint:** $$y = 2x = 2 \times \frac{1}{5} = \frac{2}{5}$$ 7. **Calculate the minimum value of $$f$$:** $$f\left(\frac{1}{5}, \frac{2}{5}\right) = \left(\frac{1}{5} - 1\right)^2 + \left(\frac{2}{5}\right)^2 = \left(-\frac{4}{5}\right)^2 + \left(\frac{2}{5}\right)^2 = \frac{16}{25} + \frac{4}{25} = \frac{20}{25} = \frac{4}{5}$$ **Final answer:** The minimum value of $$f(x,y)$$ subject to the constraint is $$\frac{4}{5}$$ at $$\left(\frac{1}{5}, \frac{2}{5}\right)$$.