Chapter 13 - Partial Derivatives - 13.9 Lagrange Multipliers - Exercises Set 13.9 - Page 996: 16

$f$ is tangent to the constraint curve $0=g(x, y)$.

Given: $f(x, y)$ is subject to the constraint $0=g(x, y)$. According to the Lagrange multipliers, the extrema value of $f(x, y)$ occurs where the constraint curve just touches a level curve. So, the maximum value will occur where $f$ is tangent to the constraint curve $0=g(x, y)$