# What are some ways to find the minimum of an expression?

I don't get how people solve these types of questions. If you have a single variable expression then I guess you could differentiate it, but how would you find the extreme values of an expression which has more than one variable? I know this is kind of a broad question, but I have absolutely no idea how would I go solving for one if I'm presented with it. For example, how would one find the minimum of $$\frac{c}{a-b} + \frac{a}{c-b} +\frac{b}{c-a}$$ Note: I made this expression myself so I wouldn't be surprised if there isn't an answer to this.

Is there any systematic way of trying to solve for these? I know the trick where one could complete the square for a quadratic expression but that is all. Are there any general rules of thumb or some tricks that are used when going about these? I will really appreciate if someone could inform me about this.

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Lagrange multipliers? –  Adam Nov 26 '13 at 1:46

For expressions like the one you put above, however, it is often the case that they can be manipulated into forms that show bounds on the possible values. For example, $x^2 - 2xy + y^2$ may not have an obvious bound, but when expressed as $(x-y)^2$ it is clear that it is never negative (for real inputs). Looking at the limits as various variables go to $0$ or $\infty$ may help bound the possible values as well.