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

Step 1: Define the Function

The function given is the absolute value function, defined as \( f(x) = |x| \), which can be written piecewise as \( f(x) = x \) for \( x \geq 0 \) and \( f(x) = -x \) for \( x < 0 \).

Step 2: Identify the Candidate for Local Minimum

We need to check if \( x = 0 \) is a candidate for a local minimum. At \( x = 0 \), \( f(0) = |0| = 0 \).

Step 3: Evaluate the Neighborhood Around x = 0

Consider the interval around 0. For \( x > 0 \), \( f(x) = x \) which is greater than \( f(0) = 0 \). For \( x < 0 \), \( f(x) = -x \) which is also greater than \( f(0) = 0 \). Thus, \( f(x) \) has a local minimum at \( x = 0 \).

Step 4: Check Differentiability at x = 0

The derivative from the right (\( x \to 0^+ \)) is \( f'(x) = 1 \), and from the left (\( x \to 0^- \)) is \( f'(x) = -1 \). Since the one-sided derivatives are not equal, \( f(x) = |x| \) is not differentiable at \( x = 0 \).

Key Concepts

Differentiability

Differentiability is a concept in calculus that describes whether a function has a derivative everywhere within its domain. In simpler terms, a function is differentiable at a point if it has a well-defined tangent at that point. A continuous curve without any breaks, sharp corners, or cusps is a good candidate for differentiability.

For a function to be differentiable at a specific point, the left-hand and right-hand limits of the derivative must not only exist but also be equal at that point. Mathematically, this is expressed as:

• The derivative from the left \(\lim_{{h \to 0^-}} \frac{f(a + h) - f(a)}{h}\)
• The derivative from the right \(\lim_{{h \to 0^+}} \frac{f(a + h) - f(a)}{h}\)

If these two derivatives do not match, the function is not differentiable at that point. In the case of the absolute value function \( f(x) = |x| \) at \(x = 0\), we find that the derivative from the right is 1, and from the left is -1. Since the two are not equal, the function is not differentiable at \(x = 0\). This is because \(f(x) = |x|\) has a sharp corner at the origin, where it switches direction abruptly.

Piecewise Functions

A piecewise function is one that is defined by different expressions based on the input value. It may seem complex, but this approach allows us to handle functions that behave differently over different intervals. For instance, the function \( f(x) = |x| \) can be expressed as:

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

This way of defining functions helps us manage abrupt changes in behavior, like the absolute value function which reflects across the y-axis. When solving problems involving piecewise functions, it is crucial to consider each piece separately for calculating limits, derivatives, or integrals.

For the absolute value function, these separate pieces help explain why the derivative behaves differently on either side of \(x = 0\), supporting our investigation into its differentiability. By addressing both parts - \(f(x) = x\) for non-negative \(x\) and \(f(x) = -x\) for negative \(x\) - we better understand how the function forms and reacts over its domain.

Local Minimum

A local minimum is a point where a function takes the smallest value in a small surrounding interval. At this point, the function value is less than or equal to other values close by. Finding a local minimum involves checking the function behavior both to the left and the right of a given point.

For the function \( f(x) = |x| \), we can see that at \(x = 0\), the value of the function is \(0\). If you move to the right (\(x > 0\)), the function value \(f(x) = x\) increases above zero. Similarly, if you move to the left (\(x < 0\)), the function \(f(x) = -x\) also increases above zero as \(x\) becomes more negative. Both observations show that there is no value smaller than \(0\) in a small interval around \(x = 0\), confirming that a local minimum occurs there.

Recognizing local minima is essential in optimization problems, as they indicate the most significant dip in a function's graph. However, while \(x = 0\) is a local minimum for \(f(x) = |x|\), the fact that it's not smooth or differentiable should always be checked to ensure all attributes are considered accurately.