If the function $$f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$$, where $$a > 0$$, attains its local maximum and local minimum values at $$p$$ and $$q$$, respectively, such that $$p^2 = q$$, then $$f(3)$$ is equal to:

We are given $$f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$$ where $$a \gt 0$$.

Finding critical points: $$f'(x) = 6x^2 - 18ax + 12a^2 = 6(x^2 - 3ax + 2a^2) = 6(x - a)(x - 2a)$$.

So the critical points are $$x = a$$ and $$x = 2a$$.

Since the coefficient of $$x^3$$ is positive, $$f$$ has a local maximum at $$x = a$$ and a local minimum at $$x = 2a$$. Therefore $$p = a$$ and $$q = 2a$$.

Given $$p^2 = q$$: $$a^2 = 2a$$, so $$a(a - 2) = 0$$. Since $$a \gt 0$$, we get $$a = 2$$.

Now $$f(x) = 2x^3 - 18x^2 + 48x + 1$$.

$$f(3) = 2(27) - 18(9) + 48(3) + 1 = 54 - 162 + 144 + 1 = 37$$.

Hence, the correct answer is Option D.