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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

Most of the techniques you know for working with a single-variable function are generalizable to higher dimensions. For instance, slope becomes gradient, which can be followed to local minima or sometimes solved analytically for zero gradient. You may also need to examine boundary conditions for an overall minimum; if the boundary is parametric in multiple variables, a change of variables may be needed to express the value of the function while following the boundary.

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.

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