Applications Of Derivatives Question 238
Question: If the function $ f(x)=2x^{3}-9ax^{2} $ $ +12a^{2}x+1, $ where $ a>0 $ attains its maximum and minimum at p and q respectively such that $ p^{2}=q $ , then a equals
[AIEEE 2003]
Options:
A) 3
B) 1
C) 2
D) $ \frac{1}{2} $
Show Answer
Answer:
Correct Answer: C
Solution:
$ f(x)=2x^{3}-9ax^{2}+12a^{2}x+1 $
$ {f}’(x)=6x^{2}-18ax+12a^{2} $
$ {f}’’(x)=12x-18a $
For maximum and minimum ,
$ 6x^{2}-18ax+12a^{2}=0\Rightarrow x^{2}-3ax+2a^{2}=0 $
$ x=a $ or $ x=2a $ , at $ x=a $ maximum and at $ x=2a $ minimum
$ \because $ $ p^{2}=q $ ,
$ \therefore $ $ a^{2}=2a\Rightarrow a=2 $ or $ a=0 $
But $ a>0 $ , therefore $ a=2. $
BETA
