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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.

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 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.