Problem 30 - Calculus for Biology and Medicine

Question

Show that \(f(x)=\left|x^{2}-1\right|\) has local minima at \(x=1\) and \(x=-1\) but \(f(x)\) is not differentiable at \(x=1\) or \(x=-1\).

Step-by-Step Solution

Answer

The function has local minima at \( x = 1 \) and \( x = -1 \), but is not differentiable at these points due to absolute value sharp edges.

Step 1: Analyze the function

The function is given by \( f(x) = |x^2 - 1| \). This is an absolute value function, which means it can be split into two cases: \( f(x) = x^2 - 1 \) if \( x^2 - 1 \geq 0 \), and \( f(x) = -(x^2 - 1) \) if \( x^2 - 1 < 0 \). Identify the points where changes occur: \( x^2 - 1 = 0 \) or \( x = \pm 1 \).

Step 2: Consider the intervals

Consider the two critical points separately:1. For \( x < -1 \) and \( x > 1 \), \( x^2 - 1 \geq 0 \) holds, so \( f(x) = x^2 - 1 \).2. For \( -1 \leq x \leq 1 \), \( x^2 - 1 < 0 \) holds, so \( f(x) = -(x^2 - 1) \).

Step 3: Determine local minima as specific points

Since \( f(x) = x^2 - 1 \) for \( x < -1 \) and \( x > 1 \), the function increases on both sides of \( x = 1 \) and \( x = -1 \). At \( x = 1 \) and \( x = -1 \), the function evaluates to \( f(1) = 0 \) and \( f(-1) = 0 \), which are lower than nearby values on both sides. Hence, local minima occur at these points.

Step 4: Investigate differentiability

A function that includes an absolute value is not differentiable at the points where the expression inside the absolute value is zero. At \( x = 1 \) and \( x = -1 \), \( x^2 - 1 = 0 \), which means \( f(x) \) changes from \( x^2 - 1 \) to its negative form, thus introducing sharp edges at these points and making \( f(x) \) non-differentiable there.

Step 5: Conclusion of differentiability and minima

Summarizing, \( f(x) \) has local minima at \( x = 1 \) and \( x = -1 \) because the function attains values lower than neighboring points. However, \( f(x) \) is not differentiable at these points due to the presence of sharp edges caused by the absolute value.

Key Concepts

Differentiability

Differentiability is a fundamental concept in calculus that refers to the behavior of a function at a specific point. If a function is differentiable at a point, it means that the derivative exists at that point, reflecting how the function changes from an "infinitesimally small" standpoint. The derivative, essentially, is the slope of the tangent line at the function’s graph at that point.
With absolute value functions, like in the case of our exercise, differentiability can get tricky. These functions can introduce sharp edges or corners. For the function \( f(x) = |x^2 - 1| \), the points \( x = 1 \) and \( x = -1 \) are such critical points where the expression inside the absolute value, \( x^2 - 1 \), equals zero, causing the change in behavior of the function from "positive" to "negative" or vice versa.

• When approaching from different sides (left and right), the slopes are not the same.
• This results in discontinuous derivatives at these points and thus, the function is not smooth.

These sharp transitions mean the function lacks a derivative at these points. Therefore, the function is not differentiable at \( x = 1 \) and \( x = -1 \). Understanding where and why functions are not differentiable helps in analyzing their graphs and behavior.

Local Minima

Local minima occur at points in a function where the value is lesser than the values at nearby points. These are the 'valleys' in the graph of the function. In our exercise, the function \( f(x) = |x^2 - 1| \) achieves local minima at the points \( x = 1 \) and \( x = -1 \). Here is why:

• At these points, the function is equal to zero: \( f(1) = 0 \) and \( f(-1) = 0 \).
• Points very close to \( x = 1 \) and \( x = -1 \), such as \( x = 1.1 \) or \( x = -1.1 \), have values greater than zero.

Since these are the lowest values in their intervals, they confirm the occurrence of local minima.
For local minima, derivatives also play a role. The change in sign of the derivative suggests a change in the decreasing nature to increasing near these points, signifying a local minimum. However, with the absolute value function here, we quickly identify minima simply by observing the value of the function around the critical points.

Absolute Value Function

The absolute value function can be tricky but is essential in calculus for analyzing different behaviors of a function. It is defined as the function that returns only the non-negative value of a number.
For any expression \( u(x) \), the absolute value function \( |u(x)| \):

• Equals \( u(x) \) if \( u(x) \geq 0 \).
• Equals \(-u(x)\) if \( u(x) < 0 \).

In our specific function \( f(x) = |x^2 - 1| \), the splitting of the function is crucial based on the value of \( x \). At points where \( x^2 - 1 = 0 \), such as \( x = 1 \) and \( x = -1 \), the function changes its form, and this switch creates a sharp edge.
This sharp transition is why the function might not always be smooth or differentiable. Absolute value functions help us easily determine where potential changes in direction or behavior might happen, making them an interesting aspect of study.