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

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If the function f(x)=2x3−9ax2+12a2x+1 where a>0, attains it's maximum and minimum at p and q respectively such that p2=q then 'a' equals

The correct option is B 2Given, f(x)=2x3−9ax2+12a2x+1,a>0f′(x)=6x2−18ax+12a2=6(x2−3ax+2a2)=6(x−a)(x−2a)∴p=a,q=2a⇒a2=2a⇒a=0(rejected)⇒a=2

If the function $f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1$ where $a > 0$ , attains it's maximum and minimum at p and q respectively such that $p^{2} = q$ then ' $a$ ' equals

The correct option is B $2$
Given, $f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1, a > 0$
$f^{'} (x) = 6 x^{2} - 18 a x + 12 a^{2}$
$= 6 (x^{2} - 3 a x + 2 a^{2})$
$= 6 (x - a) (x - 2 a)$
$∴ p = a, q = 2 a$
$\Rightarrow a^{2} = 2 a$
$\Rightarrow a = 0 (r e j e c t e d)$
$\Rightarrow a = 2$