91Ó°ÊÓ

Critical points are the points where the first derivatives (partial derivatives in multivariable calculus) of a function are zero. These points indicate where the function could have maxima, minima, or saddle points.
For example, consider the function provided: \( f(x, y) = e^x + e^y - e^{x+y} \).
To find its critical points, we calculate the partial derivatives \( \frac{\partial f}{\partial x} = e^x - e^{x+y} \) and \( \frac{\partial f}{\partial y} = e^y - e^{x+y} \). We then set these derivatives equal to zero and solve the resulting system of equations to locate the critical points.

Partial derivatives are derivatives of functions with multiple variables, considered with respect to one variable at a time, treating other variables as constants.
To find the critical points, we compute the partial derivatives of \( f(x, y) = e^x + e^y - e^{x+y} \):
\( \frac{\partial f}{\partial x} = e^x - e^{x+y} \)
\( \frac{\partial f}{\partial y} = e^y - e^{x+y} \)
By setting these equal to zero, we solve for the values of \( x \) and \( y \) that may give maxima, minima, or saddle points.

The Hessian matrix is a square matrix of second-order partial derivatives of a function. It helps determine the convexity or concavity of the function at a given point and, in turn, identifies the nature of critical points.
For the function \( f(x, y) \), the second partial derivatives are:
\( \frac{\partial^2 f}{\partial x^2} = e^x - e^{x+y} \)
\( \frac{\partial^2 f}{\partial y^2} = e^y - e^{x+y} \)
\( \frac{\partial^2 f}{\partial x \partial y} = -e^{x+y} \)
Construct the Hessian at the critical point (0, 0):
\[ H = \begin{bmatrix} \frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \ \frac{\partial^2 f}{\partial x \partial y} & \frac{\partial^2 f}{\partial y^2} \end{bmatrix} = \begin{bmatrix} 0 & -1 \ -1 & 0 \end{bmatrix} \]

Saddle points are points on the graph of a function where the slope of the tangent plane equals zero, but the point is not a local maximum or minimum. Instead, it is a point where the function changes concavity.
In our example, the Hessian determinant at \ (0,0) \ is computed as:
\[ |H| = 0 \cdot 0 - (-1) \cdot (-1) = -1 \]
Since the determinant is negative, the critical point (0, 0) is classified as a saddle point.

A relative maximum is a point where a function reaches a higher value than at any nearby points. To determine if a critical point is a relative maximum, we use the Hessian matrix.
If the determinant of the Hessian is positive and all principal minors of the Hessian are positive, the function has a relative maximum at that point. However, this is not the case in our example, where the determinant is negative.

A relative minimum is a point where the function has a lower value than at any nearby points.
Like the relative maximum, determining if a critical point is a relative minimum involves examining the Hessian: if the determinant is positive and the principal minors alternate in sign, we have a relative minimum.
Since our function's Hessian determinant is negative at (0, 0), it does not indicate a relative minimum but a saddle point instead.