91Ó°ÊÓ

Understanding the concept of partial derivatives is crucial in multivariable calculus. They allow us to examine how a function changes as we alter each individual variable, holding the others constant.
Partial derivatives are similar to ordinary derivatives but are applied to functions with two or more variables. In our example, the function is dependent on two variables, \(x\) and \(y\). Thus, the partial derivatives \(f_x\) and \(f_y\) tell us how the function changes with respect to \(x\) and \(y\) respectively. By calculating these, we gain insight into how the function's surface tilts in each direction.
When we set these derivatives to zero, we find points that could represent peaks, valleys, or saddle points in the function's surface – points where the function does not slope in any direction.

Critical points are essential in calculus optimization since they represent spots where the function may achieve a local maximum, local minimum or a saddle point.
To find critical points, we solve the system of equations formed by setting each first partial derivative to zero.
In the provided example, by solving the system, we determine that there is a critical point at \((1, 0)\). These points are important because they potentially identify peaks or troughs on the function surface. Properly identifying and classifying critical points is the backbone of optimizing functions in calculus.

The Hessian determinant is a powerful tool used to classify critical points. This calculation helps us understand the nature of these points – whether they are local minima, local maxima, or saddle points.
The Hessian matrix is made from the second partial derivatives of a function and is used to compute the Hessian determinant, \(H\). For a function of two variables, like ours, \(H\) is calculated as \[H = f_{xx}f_{yy} - (f_{xy})^2\] where \(f_{xx}\), \(f_{yy}\), and \(f_{xy}\) are the second partial derivatives.
In our case, \(H = 4\), which indicates a positive determinant. With \(f_{xx} > 0\), we conclude that the critical point \((1, 0)\) is a local minimum. Understanding the Hessian determinant's role helps clarify why certain critical points are classified in particular ways.