Algebraic likelihood maximization avoiding the log-likelihood function and differentiation

The fact that the graph of the exponential function exp is always at or above the straight line through the origin with slope exp⁡(1) is well-known and can be easily proved using differential calculus. We provide a simple algebraic proof of that fact and use that fact to construct a template for maximizing the likelihood function that avoids the log-likelihood function and differentiation in a number of examples, including the two-parameter normal distribution family.