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

Step 1: Find the derivative of the function

To find the local maximum and minimum points of a function, we first need to find its derivative. The derivative $f'(x)$ gives us the slope of the tangent to the curve at any point $x$. We apply the power rule for differentiation to each term.

Step 2: Find critical points by setting the derivative to zero

Local maximum and minimum values occur at critical points where the derivative of the function is zero. We set $f'(x) = 0$ and simplify the quadratic equation by dividing by 6.

Step 3: Solve for x to find p and q

We solve the quadratic equation $x^2 - 3ax + 2a^2 = 0$ by factoring. The roots of this equation are the critical points where the local maximum and minimum occur. These roots are $x=a$ and $x=2a$.

Step 4: Assign p and q based on the second derivative test

To determine which critical point corresponds to the local maximum ($p$) and which to the local minimum ($q$), we use the second derivative test. We find the second derivative $f''(x)$. If $f''(x) < 0$, it's a local maximum; if $f''(x) > 0$, it's a local minimum. Since $a>0$, $f''(a) = -6a < 0$, so $p=a$. And $f''(2a) = 6a > 0$, so $q=2a$.

Step 5: Use the condition $p^2 = q$ to find the value of a

We are given the condition $p^2 = q$. Substituting the values of $p=a$ and $q=2a$ into this equation, we get $a^2 = 2a$. Solving for $a$, we find two possible values: $a=0$ or $a=2$.

Step 6: Select the correct value of a and calculate f(3)

The problem states that $a > 0$, so we discard $a=0$ and choose $a=2$. Now we substitute $a=2$ back into the original function $f(x)$ to get the specific function. Finally, we evaluate $f(3)$ by substituting $x=3$ into the function.

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