I try to find the points on the surface
$$E=\left\{(x,y,z)\in\mathbb{R}^3:xy^2z^4=\frac14\right\}$$ that are closest to the origin. I started to look for points on the surface at which the gradient is parallel to the position vector, but using the AM-GM inequality is simpler.
What I did:
$$\text{max}\{xy^2z^4\} =\frac{1}{4}x^2 +y^2 +z^2$$ then have squared both sides and stuck. What is next?