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If the function f(x) = 2x 3 – 9ax 3 + 12a 2 x + 1, where a>0, attains its max. and min. at p and q respectively such that p 2 = q then a equals (A) ½ (B) 1 (C) 2 (D) 3

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

Grade:12th pass

2 Answers

arush
19 Points
6 years ago
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
Samyak Jain
333 Points
5 years ago
f(x) = 2x3 – 9ax2 + 12a2x + 1, where a>0, attains its minimum and maximum at p and q 
respectively & p2 = q
For minimum and maximum values of f(x), we differentiate it w.r.t. x equate the result to zero.
So f ’(x) = 2(3)x2 – 9a(2)x + 12a2  =  6x– 18x + 12a= 0
i.e.  x– 3x + 2a= 0  or  (x – a)(x – 2a) = 0
=> x = a or 2a
Given :  p2 = q   =>  (a)2 = 2a  =>  a(a – 2) = 0  =>  a = 0 or 2
But it is given that a > 0
\therefore a = 2

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