91Ó°ÊÓ

In multivariable calculus, partial derivatives are essential for finding critical points of a function. For the function \( g(x, y) = 4xy - x^4 - y^4 \), partial derivatives reveal how the function changes with respect to one variable while keeping the other constant. To find these, we differentiate the function with respect to each variable separately.
\( g_x(x, y) \) represents the rate of change of the function as \( x \) varies, which simplifies to \( 4y - 4x^3 \). Similarly, \( g_y(x, y) \) represents the rate of change as \( y \) changes, simplifying to \( 4x - 4y^3 \).
By setting these partial derivatives to zero, \( g_x(x, y) = 0 \) and \( g_y(x, y) = 0 \), we find critical points where the function may have a maximum, minimum, or saddle point.

• Partial Derivative with respect to \( x \): \( 4y - 4x^3 \)
• Partial Derivative with respect to \( y \): \( 4x - 4y^3 \)

Once we have located potential critical points, the second derivative test helps us classify them. In this test, we examine the second partial derivatives of the function. For \( g(x, y) \), the second partial derivatives are \( g_{xx} = -12x^2 \) and \( g_{yy} = -12y^2 \).
The second derivative test involves the Hessian determinant, which we will discuss shortly. But fundamentally, this test indicates whether a critical point is a local maximum, local minimum, or saddle point by analyzing the concavity of the function around those points. The behavior surrounding the critical points tells us which type each point is.

• Second Partial Derivative with respect to \( x \): \( g_{xx} = -12x^2 \)
• Second Partial Derivative with respect to \( y \): \( g_{yy} = -12y^2 \)

The Hessian determinant, denoted as \( H \), is crucial for the second derivative test. It's constructed from the second partial derivatives:
\[ H = g_{xx}g_{yy} - (g_{xy})^2 \]
Here, \( g_{xy} \) is the mixed partial derivative, which for this function is 4. The Hessian tells us about the nature of a critical point:
- If \( H > 0 \) and \( g_{xx} > 0 \), we have a local minimum.- If \( H > 0 \) and \( g_{xx} < 0 \), it's a local maximum.- If \( H < 0 \), the critical point is a saddle point.- If \( H = 0 \), the test is inconclusive.Calculating the Hessian at each critical point gives us clarity on their types.

Finally, classifying critical points is the key outcome of our computations. We apply the second derivative test and the Hessian determinant to each critical point to determine its nature.
For \((x, y)\):

• At \((0,0)\), \( H = -16 \), indicating a saddle point.
• At \((1,1)\), \( H = 128 \), suggesting a local maximum because \( g_{xx} < 0 \).
• At \((-1,-1)\), also \( H = 128 \), again showing a local maximum.

By identifying these classifications, we deepen our understanding of the function's geometrical nature, showing where it rises, falls, or changes direction.