Maximization of a weighted average

Question

Suppose that $f$ is a weighted average of the functions $g_i$, $$f(x) = \sum_{i=1}^N \lambda_i \cdot g_i(x)$$ (where each $\lambda_i >0$). Under what conditions is the maximum value of $f$ a weighted average of the maximum values of the $g_i$? $$ \max_x f(x) = \sum_{i=1}^N \lambda_i \cdot \max_x g_i(x)$$