Problem 134 - Calculus Volume 1

Question

For the following exercises, find the local and absolute minima and maxima for the functions over \((-\infty, \infty)\) $$y=\frac{x^{2}-1}{x-1}$$

Step-by-Step Solution

Answer

The function has no local or absolute minima or maxima over \((-\infty, \infty)\).

Step 1: Simplify the Function

The given function is \[ y = \frac{x^2-1}{x-1} \].Notice that the numerator can be factored as \[ x^2 - 1 = (x-1)(x+1) \].Cancel the common term \((x-1)\) in the numerator and denominator, which gives:\[ y = x+1 \] for \( x eq 1 \). This simplifies to a line with a hole at \(x = 1\).

Step 2: Determine the Domain

The domain of \( y = \frac{x^2-1}{x-1} \) is all real numbers except \( x = 1 \) because division by zero is undefined there.

Step 3: Assess the Critical Points

Since \( y = x+1 \) is a straight line and has no critical points in its domain (as its derivative \( y' = 1 \) is constant and non-zero), there are no local minima or maxima.

Step 4: Identify Absolute Minima and Maxima

Since the function simplifies to a straight line \( y = x + 1 \) and is defined for all \( x eq 1 \), both the local and absolute minima and maxima do not exist for the interval \((-\infty, \infty) \). The function has no bounds and extends indefinitely in both directions.

Key Concepts

local minima and maxima

Local minima and maxima refer to the lowest or highest points on a graph within a certain range or interval. For a curve, a local minimum is where the function changes from decreasing to increasing, while a local maximum is where it changes from increasing to decreasing.
In simpler terms, a local minimum is a dip, and a local maximum is a peak, but only within a specific nearby area, not necessarily the entire graph. Mathematically, these points can be identified by looking at the derivative of the function. A change in the sign of the derivative indicates a potential local minimum or maximum.

• A local minimum exists if there is a small interval around it where the function's value is less than its value at nearby points.
• A local maximum exists if the function's value at that point is greater than at nearby points within a small interval.
• For a straight line, like the simplified version of the function in this exercise, there aren't any changes in direction, so no local minimum or maximum points exist.

critical points

Critical points of a function occur where the derivative is zero or undefined. These are potential locations for local minima and maxima and are key to determining the behavior of functions.
To find critical points, you take the derivative of the function and set it to zero, solving for the variable. Additionally, you must consider any points where the derivative does not exist, as these points might also affect the graph's behavior.

• In this exercise, the derivative simplifies to a constant, meaning it never actually reaches zero or changes. Thus, there are no critical points.
• If the function had a more complex form, assessing where its derivative equals zero or is undefined would help identify potential points for further analysis.

domain of a function

The domain of a function is the complete set of possible values of the independent variable, usually denoted as x, for which the function is defined.
In practical terms, if substituting a value of x into an equation gives a valid, real number output, that x is in the domain. Conversely, any x that results in division by zero or an undefined value is not in the domain.

• For the exercise's original function, the domain is all real numbers except for x = 1. This is because division by zero is undefined at x = 1.
• Even though after simplification the function looks like a straight line, it retains this hole at x = 1 due to its original form.
• Knowing the domain helps us understand where a function is safely usable and avoids mathematical pitfalls.