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A differentiable function is a smooth, continuous function that has a derivative at every point in its domain. This essentially means that the function has no sharp corners or abrupt changes in direction. In mathematical terms, if a function is differentiable at a point, the rate of change of the function at that point is well-defined. Differentiability is a key concept in calculus because it allows us to use derivatives to analyze and predict the behavior of functions.
In the context of multi-variable functions like those in our saddle point problem, differentiability implies that we can take partial derivatives with respect to each variable. For instance, for a function \(f(x, y)\), differentiability allows us to compute \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\). Understanding how a function changes in response to small changes in \(x\) and \(y\) is crucial to finding critical points and identifying features like saddle points.

Partial derivatives are derivatives of functions of multiple variables with respect to one variable while holding the other variables constant. This is crucial when analyzing how a function behaves in different directions at any given point.

• For instance, consider the function \(f(x, y)\). The partial derivative of \(f\) with respect to \(x\) is given by \(\frac{\partial f}{\partial x}\), which represents how \(f\) changes as \(x\) changes, while \(y\) is kept constant.
• Similarly, \(\frac{\partial f}{\partial y}\) tells us how \(f\) changes as \(y\) changes, with \(x\) held constant.

Calculating partial derivatives is a crucial step in identifying critical points, which are candidates for local maxima, minima, or saddle points.
In our exercise, we first found the partial derivatives of each function and solved them to zero to find the critical points. This method helps in detecting where the function doesn't have a clear increasing or decreasing tendency, a potential indicator of a saddle point.

The second derivative test is a method to classify critical points of differentiable functions of several variables. It helps us determine whether a given critical point is a local maximum, local minimum, or saddle point.

• First, calculate the second partial derivatives: \(f_{xx}\), \(f_{yy}\), and \(f_{xy}\). These represent how the rates of change in one direction change as you move in other directions.
• The discriminant, denoted as \(D\), is computed as \(D = f_{xx}f_{yy} - (f_{xy})^2\).

If \(D > 0\), and \(f_{xx} > 0\), the point is a local minimum. If \(D > 0\), and \(f_{xx} < 0\), it's a local maximum. But if \(D < 0\), the point is a saddle point. When \(D = 0\), the test is inconclusive. This test was applied in our exercise to distinguish between saddle points and other critical points for the given functions.