If the functions $$f(x) = \frac{x^3}{3} + 2bx + \frac{ax^2}{2}$$ and $$g(x) = \frac{x^3}{3} + ax + bx^2$$, $$a \neq 2b$$ have a common extreme point, then $$a + 2b + 7$$ is equal to

We need to find $$a + 2b + 7$$ given that $$f(x)$$ and $$g(x)$$ have a common extreme point.

$$f'(x) = x^2 + ax + 2b$$

$$g'(x) = x^2 + 2bx + a$$

$$f'(c) = c^2 + ac + 2b = 0 \quad \text{...(i)}$$

$$g'(c) = c^2 + 2bc + a = 0 \quad \text{...(ii)}$$

$$(a - 2b)c + (2b - a) = 0$$

$$(a - 2b)(c - 1) = 0$$

Since $$a \neq 2b$$, we must have $$c = 1$$.

$$1 + a + 2b = 0 \implies a + 2b = -1$$

$$a + 2b + 7 = -1 + 7 = 6$$

The correct answer is Option D: $$6$$.