Critical points definition In this section, we will learn how to use partial derivatives to solve optimizationproblems. For today, we will examine the function of two variables. However, thisconcept can be applied to any number of variables. In 10 years, when you have a real job, your job might be to minimize the cost ofsomething or to maximize the profit of something. The function that you will strive tominimize or maximize will depend on several variables. If you have a function of onevariable, you know that to find its minimum or its maximum by setting the derivativeequal to zero and then looking at what happens to the function. When the partial derivative of f with respect to x is equal to 0 and the partialderivative of f with respect to y is also equal to 0, then this indicates that f(x,y) has alocal minimum or a local maximum. So why is that? Well, let's say that f(x) = 0 when x varies. That means when x varies,f does not change. Maybe this is because the function goes through the minimumpoint on its graph. If we only look at the slice parallel to the x-axis, then maybe itgoes through that minimum point. If partial f partial y is not 0, then actual maximum and minimum values cannot befound by simply changing y. But if we allow ourselves to change y as long as partial fpartial y remains 0, we will find a maximum or minimum value. This can be provenmathematically by considering that f(x,y) = 0 at the location where it attains amaximum or minimum value.
The reason that this is enough is because the first-order derivative tells us that ifboth x and y are 0, then the function doesn't change at all. And of course there willbe quadratic or higher-order terms that can make the derivative more complicatedthan just a constant. The condition that the tangent plane to the graph is horizontal means that you expectto have the maximum or minimum value. And that's what you want to have. Say you have a minimum and a maximum. If thetangent plane at this point at the bottom of the graph is horizontal, then that equationbecomes z equals constant. In other words, it becomes a horizontal plane.Because of the way partial derivatives simplify, there will be points where the partialderivative with respect to x and the partial derivative with respect to y are both 0. Wesay that x0, y0 is a critical point of f if both of these conditions are met.
In general, to ensure that the partial derivatives are zero for all variables, it isnecessary to have enough constraints.