Problem 28 - Calculus for Biology and Medicine

Question

Show that \(f(x)=|x-1|\) has a local minimum at \(x=1\) but \(f(x)\) is not differentiable at \(x=1\).

Step-by-Step Solution

Answer

\( f(x) \) has a local minimum at \( x=1 \) but is not differentiable there.

Step 1: Understand the Function

The function given is \( f(x) = |x-1| \). This is an absolute value function, which means it creates a V-shape graph where the vertex is located at the point where the inside expression is zero.

Step 2: Determine the Vertex

Set \( x-1 = 0 \) to find the x-coordinate of the vertex. Solving gives \( x = 1 \). At \( x = 1 \), \( f(x) = |1-1| = 0 \). Thus, the vertex, or the point of interest, is at (1, 0).

Step 3: Analyze the Graph Shape

Since an absolute value function \( |x-c| \) forms a V-shape, the vertex at \( x = 1 \) is the lowest point. As such, \( f(x) \) has a local minimum at \( x = 1 \).

Step 4: Check Differentiability

For differentiability at \( x = 1 \), check the left and right-hand derivatives. For \( x < 1 \), \( f(x) = -(x-1) = 1-x \), and its derivative is \( f'(x) = -1 \). For \( x > 1 \), \( f(x) = x-1 \), and its derivative is \( f'(x) = 1 \). These one-sided derivatives (-1 and 1) are not equal at \( x = 1 \), thus \( f(x) \) is not differentiable there.

Key Concepts

Local Minimum

An absolute value function like \( f(x) = |x - 1| \) often features a local minimum due to its V-shaped graph. In mathematics, a local minimum is a point where the function value is lower than at any other nearby points. For the function \( f(x) = |x - 1| \), the local minimum occurs at the vertex of the V-shape. This is where the expression inside the absolute value equals zero, which we find by setting \( x - 1 = 0 \), resulting in \( x = 1 \).

At \( x = 1 \), the value of the function is minimized to \( f(1) = |1-1| = 0 \). This tells us that the point \( (1, 0) \) is the lowest point on the graph, indicating a local minimum. This attribute of absolute value functions ensures that their vertices often serve as local minima, making them straightforward to identify.

In practical terms, identifying a local minimum is essential when analyzing trends or optimizing functions in various applications like physics or economics. It represents a stable point where small changes to input \( x \) result in no decrease in the value of \( f(x) \).

Differentiability

Differentiability refers to the ability of a function to have a derivative at a particular point. For the absolute value function \( f(x) = |x - 1| \), checking differentiability at \( x = 1 \) involves looking at the behavior of the slope on either side of this point.

On each side of \( x = 1 \), the function behaves differently due to the nature of absolute values:

• For \( x < 1 \), the function is \( f(x) = -(x-1) = 1 - x \), with a derivative \( f'(x) = -1 \).
• For \( x > 1 \), the function is \( f(x) = x-1 \), with a derivative \( f'(x) = 1 \).

The derivatives \( -1 \) and \( 1 \) are not equal, illustrating that there is a jump, or cusp, at \( x = 1 \). This difference means the function does not have a well-defined tangent line at that point, thus \( f(x) \) is not differentiable at \( x = 1 \).

Understanding differentiability helps in determining where functions are smooth and continuous vs. where they have sharp turns or corners. Such insights are valuable for simulations and modeling scenarios where smoothness impacts outcomes.

Vertex

In the context of the absolute value function \( f(x) = |x-1| \), the vertex is the critical point where the function's direction changes, presenting the V-shape. The vertex for this function is found by determining where the inside expression \( x - 1 \) becomes zero. By solving \( x - 1 = 0 \), the x-coordinate of the vertex is discovered to be \( x = 1 \).

At \( x = 1 \), the function value is minimized, resulting in the vertex point \( (1, 0) \). The vertex effectively indicates the transition from a decreasing to an increasing function as we move from left to right on the graph. This is why the vertex often corresponds to the local minimum for absolute value functions. It's the "turning point" of the graph.

The concept of a vertex extends beyond just visuals; it is crucial in solving real-world problems involving optimization. By identifying the vertex, one can quickly determine points of lowest or highest function values, useful for evaluating minimum costs or maximum efficiency in engineering or economics.