Multivariable Utility functions For a highschool economics/mathematics interdisciplinary essay I will use the Lagrange multipliers and deriving formulas that find the maximum. Could some one maybe suggest any (utility) functions that have 3 (also 4) variables (doesn't matter if it has constants).
Thanks in advance!
Let's say you have $k$ goods $x_1, x_2, \ldots, x_k$. Let $\mathbf{x}$ denote the vector $\mathbf{x} = \begin{bmatrix}x_1 \\ \ldots \\  x_k \end{bmatrix}$.

Some common workhorses to get you started:


Multivariate Cobb-Douglas: $u(\mathbf{x}) = \prod_{i=1}^k x_i^{\lambda_i} $. 


Note that $\max_\mathbf{x} u(\mathbf{x})$ has the same solution as $\max_\mathbf{x} \log u(\mathbf{x})$ because the logarithm is a monotonically increasing function. Hence maximizing the utility function $u(\mathbf{x}) = \sum_{i=1}^k \lambda_i \log x_i$ will give the same solution and also is referred to as Cobb-Douglas.

Constant elasticity of substitution $u(\mathbf{x}) = \left( \sum_i \alpha_i ^\frac{1}{s} x_i ^\frac{s - 1}{s}\right)^\frac{s}{s-1}$