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

Solution :
$$\eqalign{ & f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1 \cr & f'\left( x \right) = 6{x^2} - 18ax + 12{a^2};f''\left( x \right) = 12x - 18a \cr & {\text{For max}}{\text{. or min}}{\text{.}} \cr & 6{x^2} - 18ax + 12{a^2} = 0 \Rightarrow {x^2} - 3ax + 2{a^2} = 0 \cr & \Rightarrow x = a\,{\text{or}}\,x = 2a.\,{\text{At}}\,x = a\max .\,{\text{and}}\,{\text{at}}\,x = 2a\min \cr & \therefore p = a\,{\text{and}}\,q = 2a \cr & {\text{As}}\,{\text{per}}\,{\text{question}}\,{p^2} = q \cr & \therefore {a^2} = 2a \Rightarrow a = 2\,{\text{or}}\,a = 0 \cr & {\text{but}}\,a > 0,\,{\text{therefore}},\,a = 2. \cr} $$