Question

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:

• A. 37
• B. 40
• C. 43
• D. 46

Answer 1

Answer: (A)

37

Answer 2

The first derivative of $f(x)$ is $f'(x) = 6x^2 - 18ax + 12a^2$. Setting $f'(x) = 0$ gives $x^2 - 3ax + 2a^2 = 0$, which factors as $(x-a)(x-2a)=0$. The critical points are $x=a$ and $x=2a$. The second derivative is $f''(x) = 12x - 18a$. For $x=a$, $f''(a) = 12a - 18a = -6a$. Since $a>0$, $f''(a)<0$, so $x=a$ is a local maximum. Thus, $p=a$. For $x=2a$, $f''(2a) = 12(2a) - 18a = 24a - 18a = 6a$. Since $a>0$, $f''(2a)>0$, so $x=2a$ is a local minimum. Thus, $q=2a$. Given $p^2 = q$, we have $a^2 = 2a$. Since $a>0$, we can divide by $a$ to get $a=2$. Now, substitute $a=2$ into $f(x)$: $f(x) = 2x^3 - 9(2)x^2 + 12(2^2)x + 1 = 2x^3 - 18x^2 + 48x + 1$. Finally, evaluate $f(3)$: $f(3) = 2(3)^3 - 18(3)^2 + 48(3) + 1 = 2(27) - 18(9) + 144 + 1 = 54 - 162 + 144 + 1 = 37$.