1. **State the problem:** We are given the function $$f(x) = -2x^2 + 10x + 1$$ and the set $$X = \{1, 3, 5, 7, 9\}$$. We need to find the maximum value of $$f(x)$$ for $$x \in X$$. 2. **Formula and approach:** Since $$f(x)$$ is a quadratic function, we can evaluate $$f(x)$$ at each point in $$X$$ and find the maximum value. 3. **Calculate $$f(x)$$ for each $$x$$ in $$X$$:** - For $$x=1$$: $$f(1) = -2(1)^2 + 10(1) + 1 = -2 + 10 + 1 = 9$$ - For $$x=3$$: $$f(3) = -2(3)^2 + 10(3) + 1 = -18 + 30 + 1 = 13$$ - For $$x=5$$: $$f(5) = -2(5)^2 + 10(5) + 1 = -50 + 50 + 1 = 1$$ - For $$x=7$$: $$f(7) = -2(7)^2 + 10(7) + 1 = -98 + 70 + 1 = -27$$ - For $$x=9$$: $$f(9) = -2(9)^2 + 10(9) + 1 = -162 + 90 + 1 = -71$$ 4. **Find the maximum value:** Among the values $$9, 13, 1, -27, -71$$, the maximum is $$13$$ at $$x=3$$. **Final answer:** The maximum value of $$f(x)$$ for $$x \in X$$ is $$\boxed{13}$$ at $$x=3$$.