Question

If the function $f(x)=2x^3-9a{x}^2+12a^{2}x+1$ attains its maximum and minimum at $p$ and $q$ respectively such that $p^2=q$ , then $a$ equals

• A. $0$
• B. $1$
• C. $2$
• D. None of the

Answer 1

Answer: (C)

Given, $f(x)=2x^3-9a{x}^2+12a^{2}x+1$ attains
maximum and minimum at $p$ and $q$ respectively.
$\therefore f^{\prime }(p)=0,f^{\prime }(q)=0$
$f^{\prime \prime }(p)<0$ and $f^{\prime \prime }(q)>0$
Now$f^{\prime }(p)=0$
and $f^{\prime }(q)=0$
$\Rightarrow 6p^2-18ap+12a^2=0$
and $6q^2-18aq+12a^2=0$
$\Rightarrow p^2-3ap+2a^2=0$
and $q^2-3aq+2a^2=0$
$\Rightarrow p=a,2a,q=a,2a...(i)$
Now,$f^{\prime \prime }(p)<0$
$\Rightarrow 12p-18a<0$
$\Rightarrow p<\frac{3}{2}a...(ii)$
and $f^{\prime \prime }(q)>0\Rightarrow 12q-18a>0$
$\Rightarrow q>\frac{3}{2}a...(iii)$
and $f^{\prime \prime }(q)>0\Rightarrow 12q-18a>0$
$\Rightarrow {}^{\prime }q>\frac{3}{2}a$
From Eqs. (i), (ii) and (iii), we get $p=a,q=2a$
Now, $p^2=q$
$\Rightarrow a^2=2a$
$\Rightarrow a=0,2$
But for $a=0,f(x)=2x^3+1$ which does not
attains a maximum or minimum for any value of $x$.
Hence, $a=2$.