Question:

Step 1: Find the maximum value of $f(x)$ The function is given by $f(x) = a - 1 + 2x - x^2$. Rewrite $f(x)$ by completing the square: $$f(x) = -(x^2 - 2x) + a - 1$$ $$f(x) = -(x^2 - 2x + 1 - 1) + a - 1$$ $$f(x) = -(x - 1)^2 + 1 + a - 1$$ $$f(x) = -(x - 1)^2 + a$$ Since the coefficient of $(x - 1)^2$ is negative, the parabola opens downwards, and its maximum value occurs at $x = 1$. The maximum value of $f(x)$ is $a$. Step 2: Find the minimum value of $g(x)$ The function is given by $g(x) = x^2 - 2ax + 10 - 2a$. Rewrite $g(x)$ by completing the square: $$g(x) = (x^2 - 2ax + a^2) - a^2 + 10 - 2a$$ $$g(x) = (x - a)^2 - a^2 - 2a + 10$$ Since the coefficient of $(x - a)^2$ is positive, the parabola opens upwards, and its minimum value occurs at $x = a$. The minimum value of $g(x)$ is $-a^2 - 2a + 10$. Step 3: Apply the constraint $\max(f) < \min(g)$ The problem states that the maximum value of $f(x)$ is smaller than the minimum value of $g(x)$. $$a < -a^2 - 2a + 10$$ Rearrange the inequality to form a quadratic inequality: $$a^2 + a + 2a - 10 < 0$$ $$a^2 + 3a - 10 < 0$$ Factor the quadratic expression: $$(a + 5)(a - 2) < 0$$ This inequality holds true when $a$ is between the roots $-5$ and $2$. $$-5 < a < 2$$ Step 4: Find the largest natural number $a$ The variable $a$ must be a natural number, which means $a \in \{1, 2, 3, \dots\}$. From the inequality $-5 < a < 2$, the natural numbers that satisfy this condition are $a=1$. The largest natural number $a$ for which the condition holds is $1$.