Problem 27 - Calculus for Biology and Medicine

Question

. Show that \(f(x)=|x|\) has a local minimum at \(x=0\) but \(f(x)\) is not differentiable at \(x=0\).

Step-by-Step Solution

Answer

The function has a local minimum at \( x=0 \) but is not differentiable there due to a discontinuous derivative.

Step 1: Identify the Function Behavior

The function given is the absolute value function, \[ f(x) = |x| \]. This function is defined such that \[ f(x) = \begin{cases} -x, & \text{if } x < 0 \ x, & \text{if } x \geq 0 \end{cases} \]. The function consists of two linear pieces, \( -x \) for negative values of \( x \) and \( x \) for non-negative values of \( x \).

Step 2: Checking for Local Minimum at x=0

To determine if there is a local minimum at \( x = 0 \), observe the function values around \( x = 0 \). For \( x < 0 \), \( f(x) = -x \) which increases as \( x \) approaches 0 from the left. Similarly, for \( x \geq 0 \), \( f(x) = x \) also increases as \( x \) moves away from 0. Hence, \( f(x) \geq f(0) \) for all \( x \), and thus \( f(x) \) has a local minimum at \( x = 0 \).

Step 3: Differentiability Check at x=0

A function is differentiable at a point if the derivative exists at that point. To check if \( f(x) \) is differentiable at \( x=0 \), consider calculating the derivative from the left and right. The derivative from the left, \( x < 0 \), would be \( f'(x) = -1 \) and from the right, \( x \geq 0 \), would be \( f'(x) = 1 \). Since the left-hand derivative and the right-hand derivative at \( x = 0 \) do not agree, \( f(x) \) is not differentiable at \( x = 0 \).

Key Concepts

local minimum

In mathematics, a local minimum is a point where a function takes the smallest value in its immediate surroundings. For a function like the absolute value function, \( f(x) = |x| \), this can be easily observed around \( x = 0 \). If we evaluate \( f(x) \) at values slightly less than and greater than zero, we notice the behavior of the function.

• For \( x < 0 \), \( f(x) = -x \) reflects a positive slope approaching 0, meaning it decreases moving towards zero.
• Conversely, for \( x \geq 0 \), \( f(x) = x \), which indicates a positive slope continuing from zero as x increases.

In both intervals, the values of \( f(x) \) are higher compared to \( f(0) = 0 \). This establishes a local minimum at \( x = 0 \) because no other nearby point has a lower function value than \( f(0) \). Therefore, \( f(x) \) achieves its least value at this point, even if only locally.

absolute value function

The absolute value function, \( f(x) = |x| \), is a distinct and important mathematical concept. Its definition accommodates two situations based on whether \( x \) is negative or non-negative. Formally, it is expressed as:

• \( f(x) = -x \) when \( x < 0 \)
• \( f(x) = x \) when \( x \geq 0 \)

This results in a V-shaped graph, symmetric about the \( y \)-axis. The point at the origin \((0,0)\) is where the two linear halves of the function meet. Beyond its graph's unique shape, the absolute value function serves as a model to measure distances without considering direction and is widely used across various fields. Hence, understanding this function aids in grasping concepts of symmetry, optimization, and distance measurements.

differentiability criteria

Differentiability is a measure of a function's smoothness at a point. For a function to be differentiable at a particular point, the left-hand derivative and right-hand derivative at this point must be equal. The absolute value function \( f(x) = |x| \) , however, displays non-differentiability at \( x = 0 \) due to differing derivatives from either side:

• When approaching 0 from the left \( (x < 0) \), the derivative is \( f'(x) = -1 \).
• When approaching 0 from the right \( (x \geq 0) \), the derivative is \( f'(x) = 1 \).

These differing slopes indicate a sharp corner at \( x = 0 \). A function cannot possess a well-defined tangent line because the rates of change (slopes) differ significantly before and after \( x = 0 \). As a result, \( f(x) = |x| \) is not differentiable at this point. Understanding differentiability provides insights into the smoothness and continuity of a graph, which are pivotal in calculus and practical applications.