Maximize a linear function ● A linear function is a function that can be written as a linear equation, which means that the graph of the function is a straight line. ● A linear function is also called an affine function. Thus, the gradient vector points in a perpendicular direction relative to the level curves. The gradient of a vector has both direction and magnitude, and the magnitude of thevector's gradient is also valuable. But what is the magnitude of the gradient? So to talk about that, you can look at another warm-up problem about functions of the formAx + By + C. Imagine you have a linear function L of (x, y), which equals 3x plus 4y. Consider a simple example in which you start at 0, and then you can move in any directionby distance 1. You can pick any direction and move a distance 1. How can we maximizeL? Therefore, here is a picture to illustrate the problem.
We can start with the number 0, and then take a distance of 1 from the origin. This will giveus a point on our graph, which can be anywhere on the blue circle whose radius is 1. We would like to choose a direction so that the function L is as large as possible. The other thing you see in the picture is the level curves of the function L.