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In multivariable calculus, finding critical points is a crucial step in understanding where a function may have peaks, valleys, or flat areas. These points occur where the function's rate of change with respect to all its variables is zero. In simpler terms, we're looking at where the function stops growing or shrinking. To find these points, calculate the partial derivatives of the function with respect to each variable.

• Take each partial derivative and set them equal to zero.
• The resulting system of equations will help identify potential critical points.

As in our given function, the critical point was found by solving the equations derived from setting \(f_{x}=0\) and \(f_{y}=0\) yielding the point \(\left(\frac{2}{3}, \frac{4}{3}\right)\). This makes it a strong tool for pinpointing where a function's behavior changes.

Partial derivatives are fundamental in multivariable calculus as they extend the concept of a derivative to functions of several variables. They measure how the function changes as you vary one of the variables, while keeping others constant. This is particularly useful for functions defined on multiple dimensions. For example, for a function \(f(x, y)\), the partial derivative with respect to \(x\) is \(\frac{\partial f}{\partial x}\), and with respect to \(y\) is \(\frac{\partial f}{\partial y}\).

• To find the partial derivative with respect to \(x\), treat \(y\) as a constant, and vice-versa when finding with respect to \(y\).
• These derivatives highlight how changes in one variable affect the function's output.

In our function:\(f_{x} = 2y - 10x + 4\) and \(f_{y} = 2x - 4y + 4\), these indicated where the function's slope is zero in each direction.

To understand the nature of critical points, the second derivative test is performed. This test helps classify whether each critical point is a local minimum, local maximum, or a saddle point. For a function of two variables, the formula involves the Hessian matrix determinant.

• Compute the second derivatives \(f_{xx}, f_{yy},\) and the mixed partial derivative \(f_{xy}\).
• The determinant of the Hessian matrix is given by \(D = f_{xx}f_{yy} - (f_{xy})^2\).

The signs of \(D\) and \(f_{xx}\) help determine:- If \(D > 0\) and \(f_{xx} > 0\), then we have a local minimum.- If \(D > 0\) and \(f_{xx} < 0\), then a local maximum occurs.- If \(D < 0\), the point is a saddle point.In our exercise, \(D = 36\) and \(f_{xx} = -10\), indicating a local maximum at \(\left(\frac{2}{3}, \frac{4}{3}\right)\).

The Hessian matrix is an essential tool in multivariable calculus for evaluating the second derivatives of a function. It lays the groundwork for determining the nature of critical points. The matrix is square, containing all the second-order partial derivatives of the function at a point.

• The Hessian of a function \(f(x, y)\) is:\[H = \begin{pmatrix} f_{xx} & f_{xy} \ f_{yx} & f_{yy} \end{pmatrix}\]
• The properties of this matrix help identify the specific characteristics of each critical point.

To apply this matrix:- Compute \(H\) at the critical point.- Use its determinant in the second derivative test.In our context, \(f_{xx} = -10\), \(f_{yy} = -4\), and \(f_{xy} = 2\) contributed to outlining the function's behavior at the critical point.