This is a somewhat vague and philosophical question.
Consider the following twothree 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 $f(x^*) = a\sqrt{\frac ba} + \frac{b}{\sqrt{\frac ba}} = 2\sqrt{ab}~.$
Problem 3:
Minimize over $x> 0,$ the function $f(x) = \sqrt{a^2+x^2}-bx$ where $a>0, 0<b<1.$
Ans: $x^* = \frac{a}{\sqrt{1-b^2}}$ and $f(x^*) = \sqrt{a^2 + \frac{a^2}{1-b^2}} - b\frac{a}{\sqrt{1-b^2}} = \frac{a(\sqrt{2-b^2}-b)}{\sqrt{1-b^2}}~.$
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!