2015 NS 28

y =    8x^3  + 36x  + 54x^(-1) + 27x^(-3)        take the derivative....set to 0

y '  =  24x^2 + 36 - 54x^(-2) - 81x^(-4)  = 0   

y ' = 3 ( 8x + 12 - 18x^(-2) - 27x^(-4)  )  =  0

8x^2 + 12 - 18x^(-2) - 27x^(-4) = 0

8x^6 + 12x^4 - 18x^2 - 27 = 0   factor

4x^4 ( 2x^2 + 3)  -  9(2x^2 + 3)  = 0

(4x^4 - 9) (2x^2 + 3) = 0      the second factor gives a non-real answer when set to 0

So

4x^4 - 9 = 0

4x^4 = 9

x^4 = 9/4

x =  (9/4)^(1/4)  =  [(3/2)^2] ^(1/4)  = √[3/2]

Subbing this back into the original function

8[√[3/2] ]^3 + 36(√[3/2] ) + 54 / (√[3/2]) + 27 / [ √[3/2] ]^3   =

8 (3/2)(√[3/2] )  + 18√6   +     54 ( √[ 3/2] ) / (3/2)   +  27 /  [ (3/2) √[3/2] )

12√[3/2]   + 18√6 +  + 36  √[3/2]  +   18 / √[3/2]

6√6  + 18√6  + 18√6   +  6√6  =

48√6