If 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 equals
Options:
A .  
3
B .  
1
C .  
2
D .  
12
Answer: Option C : C We have, f(x)=2x3−9ax2+12a2x+1∴f(x)=6x2−18ax+12a2=0⇒6[x2−3ax+2a2]=0⇒x2−3ax+2a2=0⇒x2−2ax−ax+2a2=0⇒x(x−2a)−a(x−2a)=0⇒(x−a)(x−2a)=0⇒x=a,x=2a Now, f′(x)=12x−18a ∴f′(a)=12a−18a=−6a<0∴f(x) will be maximum at x = a i.e. p = a Also, f′(2a)=24a−18a=6a∴f(x)will be minimum at x = 2a i.e.q = 2a Given, p2=q ⇒a2=2a⇒a=2. Hence (c) is the correct answer.
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