91Ó°ÊÓ

The concept of partial derivatives is crucial when dealing with functions of multiple variables. In our given problem, the function \(f(x, y)\) involves two variables: \(x\) and \(y\). To find the critical points, we first need to compute the partial derivatives with respect to each variable.

When taking the partial derivative of \(f(x, y)\) with respect to \(x\), we treat \(y\) as a constant. The result of this differentiation gives us \(\frac{\partial f}{\partial x} = x^2 + 3y\). This expression allows us to see how the function changes as \(x\) changes, while \(y\) remains constant.

Similarly, when calculating the partial derivative with respect to \(y\), we hold \(x\) constant. This gives us \(\frac{\partial f}{\partial y} = -y^2 + 3x\). Hence, this derivative shows how the function responds to changes in \(y\), keeping \(x\) fixed.

Understanding these gradients is essential as they are used to locate points where the function can have maximum, minimum, or saddle points—collectively known as critical points.

Once we have the partial derivatives, the next step is to set them to zero because critical points occur where these derivatives are zero. This setup leads us to form a system of equations.

The equations are:

• \(x^2 + 3y = 0\)
• \(-y^2 + 3x = 0\)

These equations must be solved simultaneously to find the values of \(x\) and \(y\) that satisfy both conditions.

To solve this system, we can express one variable in terms of the other. For instance, from the second equation, \(-y^2 + 3x = 0\), we derive \(y^2 = 3x\). Substituting \(y^2\) in terms of \(x\) into the first equation helps us simplify and solve the system.

Through solving, we discover that \(x(x+9) = 0\), giving us potential solutions for \(x\) and consequently for \(y\), leading to the identification of critical points.

Multivariable calculus extends the concepts of calculus to functions with more than one variable, as seen in our function \(f(x, y)\). An integral part of multivariable calculus is exploring critical points, where the function exhibits special behaviors such as local maxima, minima, or saddle points.

Given the function \(f(x, y)\), critical points are determined by finding when the gradient (a vector comprising partial derivatives) is zero. For this, the role of partial derivatives becomes pivotal, as they construct components of the vector, \(abla f(x, y)\), indicating the direction of greatest increase of the function.

This technique is not just limited to finding critical points but extends to real-world applications, such as optimizing multivariable functions to find maximum profits or minimum costs. By understanding each variable's impact, multivariable calculus provides a robust toolset for tackling complex problems across various fields. In our function, the critical points \((0,0)\), \((-9, 3\sqrt{3})\), and \((-9, -3\sqrt{3})\) exemplify how we can analyze multivariable functions effectively using these techniques.