If the function $\mathrm{f}(\mathrm{x})=2 \mathrm{x}^2-9 a \mathrm{x}^2+12 \mathrm{a}^2 \mathrm{x}+1$, where…
If the function $\mathrm{f}(\mathrm{x})=2 \mathrm{x}^2-9 a \mathrm{x}^2+12 \mathrm{a}^2 \mathrm{x}+1$, where $\mathrm{a}>0$, attains its maximum and minimum at $\mathrm{p}$ and $\mathrm{q}$ respectively such that $\mathrm{p}^2=\mathrm{q}$, then a equals
$\frac{1}{2}$
3
1
2
Solution
$f(x)=2 x^3-9 a x^2+12 a^2 x+1$
$f^{\prime}(x)=6 x^2-18 a x+12 a^2 ; f^{\prime \prime}(x)=12 x-18 a$
For max. or min. $6 x^2-18 a x+12 a^2=0 \Rightarrow x^2-3 a x+2 a^2=0$
$\mathrm{x}=\mathrm{a}$ or $\mathrm{x}=2 \mathrm{a}$, at $\mathrm{x}=\mathrm{a}$ max.and at $\mathrm{x}=2 \mathrm{a} \min$.
$p^2=q$
$a^2=2 a \Rightarrow a=2$ or $a=0$
but $\mathrm{a}>0$, therefore, $\mathrm{a}=2$.