Critical points definition
In this section, we will learn how to use partial derivatives to solve optimization problems. For today, we will examine the function of two variables. However, this concept 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 of something or to maximize the profit of something. The function that you will strive to minimize or maximize will depend on several variables. If you have a function of one variable, you know that to find its minimum or its maximum by setting the derivative equal to zero and then looking at what happens to the function. When the partial derivative off with respect to x is equal to 0 and the partial derivative of f with respect to y is also equal to O, then this indicates that f(x,y) has a local minimum or a local maximum. An application of partial derivatives OPTIMIZATION PROBLEMS find min/max of a function f(X,y) At a local min or max f_x=0 and f_y=0 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 minimum point on its graph. If we only look at the slice parallel to the x-axis, then maybe it goes through that minimum point. If partial f partial y is not 0, then actual maximum and minimum values cannot be found by simply changing y. But if we allow ourselves to change y as long as partial f partial y remains 0, we will find a maximum or minimum value. This can be proven mathematically by considering that f(x,y) = 0 at the location where it attains : maximum or minimum value. [Graph] The reason that this is enough is because the first-order derivative tells us that if both x and y are 0, then the function doesn’t change at all. And of course there will be quadratic or higher-order terms that can make the derivative more complicated than just a constant. Approximation formula If we chang x–> x+Δx y–>y+Δy z=f(x,y)then Δz~~f_xΔx+f_yΔy The condition that the tangent plane to the graph is horizontal means that you expect to have the maximum or minimum value. And that’s what you want to have. Say you have a minimum and a maximum. If the tangent plane at this point at the bottom of the graph is horizontal, then that equation becomes z equals constant. In other words, it becomes a horizontal plane. Because of the way partial derivatives simplify, there will be points where the partial derivative with respect to x and the partial derivative with respect to y are both 0. We say that x0, y0 is a critical point off if both of these conditions are met. In general, to ensure that the partial derivatives are zero for all variables, it is necessary to have enough constraints <-> tangent plane to graph z f(x,y) is horizontal ! [Graph]
