If the function f(x) = 2x³ – 9ax³ + 12a²x + 1, where a>0, attains its max. and min. at p and q respectively such that p² = q then a equals

(A) ½ (B) 1

(C) 2 (D) 3

We have,f(x)=2x3−9mx2+12m2x+1f`(x) = 6x2−18mx+12m2For extremum value of f(x), f`(x)=0⇒6x2−18mx+12m2=0 ⇒x2−3mx+2m2=0⇒x2−2mx−mx+2m2=0⇒x(x−2m)−m(x−2m)=0⇒(x−2m)(x−m)=0⇒x=m or x=2mNow, f``(x) = 12x − 18mNow, f``(x)[at x = m] = 12m − 18m = − 6m 0So, f(x) attains maximum and minimum at m and 2m.So, m = p and 2m = qBut p2 = q⇒m2 = 2m⇒m2−2m = 0⇒m(m−2) = 0⇒m = 0 or m = 2