Question

If the function $f(x) = 2x^{3} - 9ax^{2} + 12a^{2}x + 1$, where $a > 0$

attains its maximum and minimum at p and q respectively

such that $p^{2} = q$, then a equals

• A. 3
• B. 1
• C. 2
• D. $\frac{1}{2}$

Answer 1

Answer: (C)

2

Answer 2

$f(x) = 2x^{3} - 9ax^{2} + 12a^{2}x + 1$

$f^{'}(x) = 6x^{2} - 18ax + 12a^{2}$

$f^{''}(x) = 12x - 18a$

For maximum and minimum, $6x^{2} - 18ax + 12a^{2} = 0$

⇒ $x^{2} - 3ax + 2a^{2} = 0$

$x = a$ or $x = 2a$ at $x = a$ maximum and at $x = 2a$ minimum

$\because$ $p^{2} = q$

$a^{2} = 2a$ ⇒ $a = 2$ or $a = 0$ but $a > 0,$ therefore $a = 2$.