Optimization and linear approximation To find the maximum, we will use the Boundary Value Problem solver. We will movethe cursor along the boundary until we reach an area where the value is increasingor decreasing.We started with an initial point at (2, −1) and decided that, if we went clockwise alongthe boundary, it would get bigger. So we kept going clockwise for a while. Now weare going counterclockwise back toward the starting point. Switching the presentation slides. This might seem like a cheap trick, but it is actuallyquite effective. And right now we are within this range, so we are on target. So let'ssay you're at a point (x, y) and the gradient is given by this equation. And thenconsider making a move to a nearby point, which we can get to by sliding down thegradient.The value of the function at the new point is approximately equal to the old value ofthe function plus the first derivative of x times the change in x plus the second
derivative of y times the change in y. The value of x at this new point is equal to thisold value, so this whole expression here - this is the change in f. We want to know whether the function got bigger or smaller, and so we want to knowif this change in f is positive or negative. We can write it as the dot product between(x derivative, y derivative) and (change in x, change in y).The dot product of two vectors is given by the formula (f x delta x) + (f y delta y)cos(theta), where f is any scalar function, delta x and delta y are the components ofvector x and vector y, and theta is the angle between them. You must determine the sign of the cosine of theta in order to determine whether ornot delta f is positive. If theta is less than a right angle, then the cosine of theta ispositive, which means that delta f is positive. When theta is bigger than pi over 2, it means that the cosine of theta is negative.Then, the change in f is negative.
If we look at this angle and compare it to a right angle, we can see that this angle issmaller than a right angle. This means that as we move along this line, the functionwill increase. By the time we reach this point, though, the angle is bigger than a rightangle, so if we go any farther in this direction, we will have exceeded our goal.Instead we should go back toward where we started. The Goldilocks point is when the angle of an oblique line is a right angle. If it is not ata right angle, it could go one way or the other. It would get bigger. At the maximumpoint, the gradient is normal, meaning perpendicular, to the boundary. Let's box thatand put a star next to it.