This is a somewhat vague and philosophical question.

Consider the following two problems:

Problem 1:

Minimize over all real-valued $x,$ the function $f(x) = bx-ax^2$ where $a,b>0.$

Ans: $x^* = \frac{b}{2a}$ and $f(x^*) = b\frac{b}{2a} - a\left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a}~.$

Problem 2:

Minimize over $x> 0,$ the function $f(x) = ax+\frac{b}{x}$ where $a,b>0.$

Ans: $x^* = \sqrt{\frac ba}$ and $f(x^*) = a\sqrt{\frac ba} + \frac{b}{\sqrt{\frac ba}} = 2\sqrt{ab}~.$

In both cases, we may obtain $x^*$ using differentiation, and plug it in to find that we can collect the terms in a neat way.

This is not unusual: in many (but not all) optimization problems, we find that when we compute the function at the optimizer $x^*,$ we are able to gather the different terms together in a non-clumsy way. This is especially likely to happen if the function is a sum of only two terms.

So overwhelming is my expectation, that if I don't see "nice" things happening when evaluating $f(x^*),$ I have an instinct to re-check my calculation to make sure the $x^*$ I obtained was actually right.

Has anyone else felt the same way? Why does this happen? Thanks!