As y=x^4-27x
observe that x^4-27x=x(x^3-27) and as x^4 is always positive and rises faster than 27x, curve, it is negative only between 0 < x < 3 and elsewhere it is positive.
(dy)/(dx)=4x^3-27 and (d^2y)/(dx^2)=12x^2
Now (dy)/(dx)=0 when 4x^3-27=0
i.e. (root(3)4x)^3-3^3=0
or (root(3)4x-3)((root(3)4x)^2+3root(3)4x+9)=0
or (root(3)4x-3)(root(3)16x^2+3root(3)4x+9)=0
And (dy)/(dx)=0 when root(3)4x-3=0 or x=3/root(3)4=1.8899
and as at this point (d^2y)/(dx^2)>0, we have a minima.
And for root(3)16x^2+3root(3)4x+9)=0, discriminant is 9root(3)16-4xx9root(3)16=--27root(3)16 <0 and no other solution for (dy)/(dx)=0 exists.
graph{x^4-27x [-10, 10, -60.2, 99.8]}
