If the function f(x)=2 x^3-9 x^2+12 a ^2 x+1, a >0 has a local maximum at x= and a local minimum at x= ^2 , then and ^2 are the roots of the equation :

Step-by-step solution

aligned & + ^2=3 a & ^2=2 a ^2 & & ( + ^2 )^3=27 a ^3 & 2 a ^2+4 a ^4+3(3 a ) (2 a ^2 )=27 a ^3 & 2+4 a ^2+18 a =27 a & 4 a ^2-9 a +2=0 & 4 a ^2-8 a - a +2=0 & (4 a -1)( a -2)=0 a =2 & so 6 x ^2-36 x +48=0 aligned x^2-6 x+8=0 ...(1) If we take a = 1 4 then = 1 2 which is not possible