State the Lagrangian Sufficiency Theorem for the maximization over $x$ of $f(x)$ subject to the constraint $g(x)=b$.

For each $p>0$, solve

$$\max \sum_{i=1}^{d} x_{i}^{p} \quad \text { subject to } \sum_{i=1}^{d} x_{i}=1, \quad x_{i} \geqslant 0 .$$