Bestimmen Sie dic relativen Extremwerte der folgenden Funktionen: a) \(\quad z=3 x y^{2}+4 x^{3}-3 y^{2}-12 x^{2}+1\) b) \(\quad z=\left(x^{2}+y^{2}\right) \cdot e^{-x}\) c) \(z=x y-27\left(\frac{1}{x}-\frac{1}{y}\right)\) d) \(z=\sqrt{1+x^{2}+y^{2}}\) e) \(\quad z=2 x^{3}-3 x y+3 y^{3}+1\)

When analyzing multivariable functions, partial derivatives are essential. They show how the function changes when one variable changes while the others are held constant. For a function \[ z = f(x,y) \], the partial derivatives with respect to x and y are \[ \frac{\partial z}{\partial x} \] and \[ \frac{\partial z}{\partial y} \]. Partial derivatives help us understand the function's slope in different directions.

To find a function's critical points, start by finding its first partial derivatives. For example, given \[ z = 3xy^2 + 4x^3 - 3y^2 - 12x^2 + 1 \], compute \[ \frac{\partial z}{\partial x} \] and \[ \frac{\partial z}{\partial y} \]:

\[ \frac{\partial z}{\partial x} = 3y^2 + 12x^2 - 24x \]
\[ \frac{\partial z}{\partial y} = 6xy - 6y \]

Find the partial derivatives for all functions before moving to the next steps.

Critical points are locations where the function's partial derivatives are zero. These points are candidates for relative extremum. Set each partial derivative equal to zero and solve for the variables. This creates a system of equations.

For the example above, we have:

\[ 3y^2 + 12x^2 - 24x = 0 \]
\[ 6xy - 6y = 0 \]

Solve these equations systematically to find the values of x and y. Critical points provide potential locations for maximum, minimum, or saddle points in the function.

Different functions may yield multiple or no critical points. Always verify each solution accurately.

To categorize critical points, we use the second derivative test. Compute the second partial derivatives \[ z_{xx}, z_{yy}, z_{xy} \]. These help us create the Hessian matrix, which is crucial for this test.

The Hessian matrix's determinant determines the nature of each critical point. The determinant \[ D \] for function \[ z \] is calculated as:

\[ D = z_{xx} \cdot z_{yy} - (z_{xy})^2 \]

Interpret \[ D \] at each critical point as follows:

• D > 0 and \[ z_{xx} > 0 \]: Local minimum
• D > 0 and \[ z_{xx} < 0 \]: Local maximum
• D < 0: Saddle point
• D = 0: Inconclusive

This test completes our analysis of the critical points.

The Hessian matrix is a square matrix composed of second-order partial derivatives of a function. It is crucial in the second derivative test for identifying relative extrema.

For a function \[ z = f(x,y) \], the Hessian matrix \[ H \] is:

\[ H = \begin{pmatrix} z_{xx} & z_{xy} \ \ z_{xy} & z_{yy} \ \end{pmatrix} \]

Constructing the Hessian begins with computing the second partial derivatives. Using previous partial derivatives, find the second-order terms for each variable. For the given example, the entries would be:

\[ z_{xx} = 24x - 24 \]
\[ z_{yy} = 6x - 6 \]
\[ z_{xy} = 6y \]

Compile these into the Hessian matrix, then use its determinant to evaluate critical points, streamlining what each critical point represents by referring to earlier discussed categories.