Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


g(x) = (x − 2) / (x²−1), 0 ≤ x < 1

Verified step by step guidance

1. First, find the derivative of the function g(x) = \( \frac{x - 2}{x^2 - 1} \). Use the quotient rule, which states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{u'v - uv'}{v^2} \). Here, \( u = x - 2 \) and \( v = x^2 - 1 \).
2. Calculate the derivatives: \( u' = 1 \) and \( v' = 2x \). Substitute these into the quotient rule formula to find \( g'(x) \).
3. Set the derivative \( g'(x) \) equal to zero to find critical points. Solve the equation \( \frac{(x^2 - 1) - (x - 2)(2x)}{(x^2 - 1)^2} = 0 \) to find the values of x where the derivative is zero.
4. Check the critical points and endpoints within the domain \( 0 \leq x < 1 \) to determine if they are local extrema. Evaluate the function g(x) at these points.
5. Use the first or second derivative test to classify the critical points as local minima, maxima, or neither. The first derivative test involves checking the sign of \( g'(x) \) around the critical points, while the second derivative test involves evaluating \( g''(x) \) at the critical points.

Key Concepts

Local Extrema

Local extrema refer to the points in a function where it reaches a local maximum or minimum within a specific interval. To identify these points, one typically examines the function's derivative to find critical points where the derivative is zero or undefined, and then uses the second derivative test or other methods to determine the nature of these points.

Critical Points

Critical points are values of x where the derivative of a function is either zero or undefined. These points are potential candidates for local extrema. To find them, calculate the derivative of the function and solve for x where the derivative equals zero or does not exist. Analyzing these points helps in determining the behavior of the function around them.

Domain Restrictions

Domain restrictions define the set of x-values for which a function is considered. In this problem, the domain is 0 ≤ x < 1, meaning the function is only analyzed within this interval. Understanding domain restrictions is crucial as they limit where extrema can occur and affect the behavior of the function, especially near boundaries or points of discontinuity.