Question

Describe the appearance of a smooth surface with a local maximum at a point.

Short answer

Answer: For a smooth surface function \(f(x,y)\) to have a local maximum at a point \((x_0, y_0)\), the following conditions must be met: 1. The first partial derivatives at that point, \(f_x(x_0, y_0)\) and \(f_y(x_0, y_0)\), must be zero. 2. The discriminant, \(D(x_0, y_0) = f_{xx}(x_0, y_0)f_{yy}(x_0, y_0) - (f_{xy}(x_0, y_0))^2\), should be greater than zero. 3. The second partial derivative with respect to \(x\), \(f_{xx}(x_0, y_0)\), should be negative. Visually, the smooth surface would have a concave down appearance around the local maximum point, which appears as a "peak" on the surface as it curves downwards in all directions away from the maximum point.

Step-by-step solution

Define the surface function

Let \(f(x,y)\) be a smooth surface function of two variables, \(x\) and \(y\).

Determine the first partial derivatives

Calculate the first partial derivatives of \(f(x,y)\) with respect to \(x\) and \(y\). These can be written as \(f_x(x,y)\) and \(f_y(x,y)\), respectively.

Conditions for a local maximum

For a point \((x_0, y_0)\) to be a local maximum, the first partial derivatives must be zero at that point. So, we require \(f_x(x_0, y_0) = 0\) and \(f_y(x_0, y_0) = 0\).

Determine the second partial derivatives

Calculate the second partial derivatives of \(f(x,y)\) with respect to \(x\) and \(y\). We have four possible combinations: \(f_{xx}(x,y)\), \(f_{xy}(x,y)\), \(f_{yx}(x,y)\), and \(f_{yy}(x,y)\).

Apply the second derivative test

The second derivative test helps us determine whether a point is a local maximum, local minimum, or a saddle point. Calculate the discriminant, \(D(x,y)=f_{xx}(x,y)f_{yy}(x,y)-(f_{xy}(x,y))^2\). If \(D(x_0, y_0) > 0\) and \(f_{xx}(x_0, y_0) < 0\), then there is a local maximum at \((x_0, y_0)\).

Describe the smooth surface with a local maximum

Once you have checked that the conditions in steps 3 and 5 are met, you can describe the appearance of the smooth surface. The surface should be concave down around the local maximum point, meaning that it curves downwards when moving away from the maximum point in any direction. This maximum point will be a "peak" on the surface.

Key concepts

Partial Derivatives

Partial derivatives are foundational in multivariable calculus. They measure how a function's output changes as one of the input variables is varied while the other variables are held constant. For a function defined by two variables, such as a smooth surface represented by the function f(x, y), we compute the partial derivative with respect to x to understand how the function changes along the x-axis, and a separate partial derivative with respect to y for changes along the y-axis. Concretely, the notations f_x(x, y) and f_y(x, y) represent these partial derivatives. It’s like slicing the smooth surface vertically or horizontally and analyzing the slope of the ensuing curve.

This technique is particularly powerful when looking for local maxima or minima, as these occur where the slope in all directions is zero—hence the partial derivatives equal to zero at this point. By finding where these derivatives vanish, we can locate critical points which are potential candidates for local extrema.

Second Derivative Test

Once potential locations for local extrema (where the first partial derivatives are zero) are identified, the second derivative test comes into play. This test helps distinguish whether a critical point is a local maximum, a local minimum, or a saddle point. In multivariable calculus, we consider the second partial derivatives of the function: f_{xx}(x,y), f_{yy}(x,y), and the mixed derivatives f_{xy}(x,y) and f_{yx}(x,y).

The discriminant D is calculated as D(x,y) = f_{xx}(x,y) * f_{yy}(x,y) - (f_{xy}(x,y))^2. If D at a critical point is positive and f_{xx} is negative, it indicates a local maximum because the surface is curving downwards in both the x and y directions around that point. Conversely, if f_{xx} is positive under the same conditions for D, we have a local minimum. And if D is negative, it's a saddle point—a peculiar feature where the surface is curving up in one direction and down in another.

Concavity and Points of Inflection

Concavity relates to the curvature direction of a function's graph, and this concept extends beautifully into multivariable calculus. For a smooth surface, concavity describes how the surface bends in different directions. If the surface bends downwards in all directions from a point, it is concavely shaped like an upside-down bowl, implying that point could be a local maximum. In univariate calculus, concavity is determined by the sign of the second derivative.

Points of inflection are where the concavity changes, shifting from concave up to concave down or vice versa. While these points are nuanced in multivariable functions, the concept of curvature changing direction remains crucial. Inflection points in two dimensions can be helpful to analyze changes in curvature for traces of the surface along specific planes, providing a more nuanced understanding of the surface's overall shape and behavior.

Multivariable Calculus

Multivariable calculus expands the concepts of calculus to functions of several variables. Beyond locating local maxima and minima on smooth surfaces, it encompasses a wide array of tools for analyzing and describing these surfaces. Techniques include working with vector-valued functions, partial derivatives, multiple integrals, and gradient vectors, all contributing to understanding the dynamic landscape of functions in higher dimensions.

Multivariable functions can represent real-world phenomena where multiple factors are at play, such as the temperature distribution in a room, which depends on both horizontal position and height above the floor. In such contexts, multivariable calculus allows the determination of rates of change in various directions, optimal values, and the behavior of functions across different regions of their domain – essential concepts for advanced fields like engineering, physics, and economics.