How to optimize a function of two variables?

Let’s say I have a function of two variables. If I differentiate the function respect to each variable and then set each of these equations to zero, then I have two equations I need to solve simultaneously.

However, when solving each equation for the variable in question after setting derivative to zero, the optimal estimate of that variable might be an expression that includes the other variable of the function as well.

What can I do in cases like these?

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