Verified answer

You got the wrong derivative, that is why the stationary point was wrong.

y = 4x² + 1 / x

dy / dx = 8x - 1 / x²

For stationary points, dy / dx = 0

8x - 1 / x² = 0

8x³ - 1 = 0

8x³ = 1

x³ = 1 / 8

x = 1 / 2

To the answer below, the question does not ask for the nature of the stationary point.

On the right track but made a small error of sign

dy/dx = 8x - 1/x^2 = 0

Do you remember how to check if this is a min or a max

Regards - Ian

y'=8x-x^-2=0 , 8x=x^-2 , x^3=1/8 , x=1/2

You forgot to multiply by the original power on your second term. Since 1/x is x^-1, the derivitive of that term is -x^-2