If the function [f( x ) = 2x^3 - 9ax^2 + 12a^2x + 1,] where a0, attains its maximum and m…

Question

If the function $$f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1,$$ where $a>0,$ attains its maximum and minimum at $p$ and $q$ respectively such that ${p^2} = q$ ,  then $a$ equals

Accepted Answer

$$f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1$$ $$f'\left( x \right) = 6{x^2} - 18ax + 12{a^2};$$ $$\,\,\,\,\,\,\,\,\,\,f''\left( x \right) = 12x - 18a$$ For max. or min. $6{x^2} - 18ax + 12{a^2} = 0$ $$ \Rightarrow {x^2} - 3ax + 2{a^2} = 0$$ $$ \Rightarrow x = a$$ or $x=2a.$ At $x=a$ $f''(a) = 12a - 18a = -6a < 0$ At $x=a$ maximum As f''($a$) < 0. At $x=2a$ $f''(a) = 24a - 18a = 6a > 0$ At $x=2a$ minimum As f''(2$a$) > 0. $$\therefore$$ $p=a$ and $q=2a$ As per question ${p^2} = q$ $$\therefore$$ ${a^2} = 2a$ $$ \Rightarrow $$ $a(a - 2) = 0$ $$ \therefore $$ $a$ = 0, 2 but $a > 0,$ therefore, $a=2.$

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