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Critical points are special locations on the graph of a function where the gradient is zero or undefined. For a function of two variables like \(f(x, y)\), these points occur where both of the first partial derivatives equal zero.

To find these points, you calculate the partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\). Once you've found those, set them to zero and solve the system of equations that results.

In this exercise, the critical point is found at \((b, a)\) since both \(\frac{\partial f}{\partial x} = y - a\) and \(\frac{\partial f}{\partial y} = x - b\) equals zero at this pair of \(x\) and \(y\) values.

Partial derivatives are a fundamental tool in multivariable calculus. They measure the rate of change of a function with respect to one variable while keeping others constant. For the function \(f(x, y) = x y - a x - b y\), the first partial derivatives with respect to \(x\) and \(y\) are crucial for identifying critical points.

Here, the partial derivative with respect to \(x\) is \(\frac{\partial f}{\partial x} = y - a\). This indicates how \(f\) changes as \(x\) changes, holding \(y\) constant. Similarly, the derivative \(\frac{\partial f}{\partial y} = x - b\) represents the change in \(f\) with variations in \(y\).

These derivatives help locate the critical points by setting them to zero and solving for \(x\) and \(y\).

The second derivative test helps classify critical points as local minima, local maxima, or saddle points. After finding the critical points, the next step is to examine the second partial derivatives.

For the function \(f(x, y)\), second partial derivatives are given by \(\frac{\partial^2 f}{\partial x^2}\), \(\frac{\partial^2 f}{\partial y^2}\), and the mixed partial derivative \(\frac{\partial^2 f}{\partial x\partial y}\). These are used to compute \(D(x, y)\), where:

• \(D(x, y) = \frac{\partial^2 f}{\partial x^2} \frac{\partial^2 f}{\partial y^2} - (\frac{\partial^2 f}{\partial x \partial y})^2\).

If \(D < 0\), the critical point is a saddle point. If \(D > 0\) and both second partial derivatives are positive, it is a local minimum, and if negative, a local maximum. In this function's case, \(D(b, a) = -1 < 0\), indicating a saddle point.

A saddle point is a type of critical point that is neither a local maximum nor a local minimum. Instead, it is a location on the graph where the surface has a saddle-like shape—concave in one direction and convex in the perpendicular direction.

Using the second derivative test, a saddle point is confirmed if \(D < 0\), as shown in the example problem for the function \(f(x, y)\). At the critical point \((b, a)\), the calculation of \(D(b, a) = -1\) indicates this is indeed a saddle point.

This classification means that near the point \((b, a)\), the function \(f(x, y)\) behaves differently along different paths. Some paths will show an increase while others will show a decrease, reflecting the "saddle" structure.