Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

Mean Value Theorem for Several Variables If \(f\) is differentiable at each point of the line segment from a to \(\mathbf{b}\), then there exists on that line segment a point \(\mathbf{c}\) between \(\mathbf{a}\) and \(\mathbf{b}\) such that $$ f(\mathbf{b})-f(\mathbf{a})=\nabla f(\mathbf{c}) \cdot(\mathbf{b}-\mathbf{a}) $$ Assuming that this result is true, show that, if \(f\) is differentiable on a convex set \(S\) and if \(\nabla f(\mathbf{p})=\mathbf{0}\) on \(S\), then \(f\) is constant on \(S\). Note: A set \(S\) is convex if each pair of points in \(S\) can be connected by a line segment in \(S .\)

Step-by-Step Solution

Answer

If \(\nabla f(\mathbf{p}) = \mathbf{0}\) on a convex set, then \(f\) is constant there.

Step 1: Understanding the Mean Value Theorem

The Mean Value Theorem (MVT) for several variables states that for a differentiable function on a line segment between two points \(\mathbf{a}\) and \(\mathbf{b}\), there exists a point \(\mathbf{c}\) such that \(f(\mathbf{b}) - f(\mathbf{a}) = abla f(\mathbf{c}) \cdot (\mathbf{b} - \mathbf{a})\). This implies that the change in function value is given by the gradient at some point \(\mathbf{c}\).

Step 2: Apply the Mean Value Theorem to a Convex Set

Given \(f\) is differentiable on a convex set \(S\) and \(abla f(\mathbf{p})=\mathbf{0}\) for every point \(\mathbf{p}\) in \(S\), we wish to show \(f\) is constant on \(S\). Assume two arbitrary points \(\mathbf{a}\) and \(\mathbf{b}\) in \(S\). By convexity, the line segment between them lies entirely within \(S\).

Step 3: Use the Condition of Zero Gradient

In the context of the MVT result, for any two points \(\mathbf{a}\) and \(\mathbf{b}\) in \(S\), there exists a point \(\mathbf{c}\) such that \(f(\mathbf{b}) - f(\mathbf{a}) = abla f(\mathbf{c}) \cdot (\mathbf{b} - \mathbf{a})\). Since \(abla f(\mathbf{c}) = \mathbf{0}\), the equation becomes \(f(\mathbf{b}) - f(\mathbf{a}) = \mathbf{0} \cdot (\mathbf{b} - \mathbf{a}) = 0\).

Step 4: Conclude that \(f\) is Constant

Since \(f(\mathbf{b}) - f(\mathbf{a}) = 0\), it follows that \(f(\mathbf{b}) = f(\mathbf{a})\). As \(\mathbf{a}\) and \(\mathbf{b}\) were arbitrary points in \(S\), this means the function value does not change between any two points, thus \(f\) must be constant on \(S\).

Key Concepts

differentiable function

A differentiable function is one that has a derivative at every point in its domain. Differentiability means that the function is smooth, with no sharp corners or breaks. This property is crucial when applying the Mean Value Theorem (MVT) in several variables because it ensures that we can find a gradient, a concept akin to the slope in single-variable calculus, at every point in a set. To understand this better:

• The derivative of a function gives us the rate of change or the tangent's slope at any point.
• A differentiable function allows us to compute this rate of change consistently across its domain, no sudden changes or undefined behavior.

When dealing with several variables, we're interested in knowing how the function changes when all the variables change slightly. Differentiable functions in this scenario provide a reliable way to calculate these changes, through a comprehensive tool known as the gradient, which we'll discuss further on.

convex set

A convex set is a collection of points where for any two points within the set, the line segment connecting them is entirely contained within the set. This property is essential when discussing functions defined over multiple variables because it simplifies assumptions about the shape and behavior of the function on that set.Here's why convexity matters:

• It ensures that between any two points, \( \mathbf{a} \) and \( \mathbf{b} \), within the set, all intermediate points lie within the domain of the function. This is required for evaluating functions that are differentiable across an entire region.
• Convex sets make it manageable to apply the MVT to any two points within the set, guaranteeing that the conditions of differentiability apply throughout the entire connection between those points.

This property plays a major role in our exercise because it allows us to apply the Mean Value Theorem repeatedly as part of demonstrating that a function with zero gradient is constant on such a set.

gradient

Imagine wanting to know not just how a function changes in a single direction, but in all directions from a point. The gradient is the tool that gives us this knowledge. It is a vector that embodies the rate and direction of the steepest increase of a function.Breaking it down:

• The gradient vector \( abla f \) of a function \( f \) contains all the partial derivatives of \( f \) with respect to each variable. These partial derivatives give us the rate of change of \( f \) along each axis of the coordinate system.
• In geometric terms, the direction of the gradient vector is the direction in which the function increases the most.

In the exercise, understanding the gradient is crucial, since the condition \( abla f(\mathbf{p}) = \mathbf{0} \) indicates that there is no increase in the function value, no matter which direction you take from any point \( \mathbf{p} \). This zero-gradient situation contributes directly to showing why the function remains constant.

constant function

A constant function is one whose output value never changes regardless of the input. No matter which points you choose in its domain, it always delivers the same value. This might remind you of a flat horizontal line in a graph, as there are no peaks or valleys to alter the height. Characteristics of a constant function include:

• It has a derivative of zero at every point on its domain, reflecting an absence of change or slope.
• In the context of our exercise, when the gradient is consistently zero over a convex set, this implies the function is constant throughout that set.

Recognizing when a function is constant, especially in several-variable calculus, requires understanding not just its behavior at a single point but across an entire region or set. In our example, the condition of the zero gradient over a convex set perfectly illustrates that such a function must be constant across that entire set.