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

The correct option is D 2f′(x)=6x2−18ax+12a2=6(x2−3ax+2a2)=6(x−a)(x−2a)f′(x)>0 If x>2a or x<af′(x)<0 If xϵ(a,2a)p=a q=2ap2=qa2=2aa=2

lf 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 =

The correct option is D 2
$f^{'} (x) = 6 x^{2} - 18 a x + 12 a^{2}$
$= 6 (x^{2} - 3 a x + 2 a^{2})$
$= 6 (x - a) (x - 2 a)$
$f^{'} (x) > 0$ If $x > 2 a$ or $x < a$
$f^{'} (x) < 0$ If $x ϵ (a, 2 a)$
$p = a$ $q = 2 a$
$p^{2} = q$
$a^{2} = 2 a$
$a = 2$