Find a Lagrange multipliers problem on a sphere in $\mathbb{R}^d$ which proves the inequality of arithmetic and geometric means: $$ (x^1 \cdot x^2 \cdot \cdot \cdot x^d)^{\frac{1}{d}} \leq \frac{x^1+...+x^d}{d}$$
My idea was:
Let $y^1, ..., y^d \in \mathbb{R}$.
And the sphere which contains the point defined by them $S = \left \{ \vec{x} \in \mathbb{R}^d : g(\vec{x})=||\vec{x}|| = ||(y^1, ..., y^d)||=n \right \} $.
And the function $f(x^1, ..., x^d) = x^1 \cdot \cdot \cdot x^d - (\frac{x^1+...+x^d}{d})^d$.
If we can show that $f \leq 0$ that would suffice.
But when I tried solving the equations we get from $\nabla f(\vec{x}) = \lambda \nabla g(\vec{x}), g(\vec{x}) = n$ I couldn't find a way to express the ext points, though we know they exists since $S$ is compact and $f$ is continuous.
Should I approach this in a different manner, or am I just missing how to solve these equations?
Note: there are some similar questions on the site but as far as I could find their instructions are at least somewhat different, so please make sure before offering to close this question.