First Derivative Test


a. Locate the critical points of f.
b. Use the First Derivative Test to locate the local maximum and minimum values.
c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).


f(x) = x²/(x² - 1) on [-4,4]

Verified step by step guidance

1. To find the critical points of the function \( f(x) = \frac{x^2}{x^2 - 1} \), first compute the derivative \( f'(x) \). Use the quotient rule: \( f'(x) = \frac{(x^2 - 1)(2x) - x^2(2x)}{(x^2 - 1)^2} \). Simplify the expression to find \( f'(x) \).
2. Set \( f'(x) = 0 \) to find the critical points. Solve the equation \( (x^2 - 1)(2x) - x^2(2x) = 0 \) to find the values of \( x \) where the derivative is zero. Also, consider where the derivative is undefined, which occurs when the denominator \( (x^2 - 1)^2 = 0 \).
3. Use the First Derivative Test to determine the nature of each critical point. Analyze the sign of \( f'(x) \) on intervals around each critical point to determine if the function is increasing or decreasing, which will help identify local maxima and minima.
4. To find the absolute maximum and minimum values on the interval \([-4, 4]\), evaluate \( f(x) \) at the critical points found within the interval and at the endpoints \( x = -4 \) and \( x = 4 \). Compare these values to determine the absolute extrema.
5. Consider the behavior of \( f(x) \) as \( x \) approaches the points where the function is undefined (i.e., \( x = \pm 1 \)) to ensure these do not affect the determination of absolute extrema on the closed interval.

Key Concepts

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are essential for identifying local maxima and minima, as they represent potential locations where the function's behavior changes. To find critical points, one must first compute the derivative of the function and solve for values of x that satisfy the derivative equation.

First Derivative Test

The First Derivative Test is a method used to determine whether a critical point is a local maximum, local minimum, or neither. By analyzing the sign of the derivative before and after the critical point, one can conclude that if the derivative changes from positive to negative, a local maximum exists; if it changes from negative to positive, a local minimum is present.

Absolute Maximum and Minimum

The absolute maximum and minimum values of a function on a closed interval are the highest and lowest values the function attains within that interval, including at the endpoints. To find these values, one must evaluate the function at the critical points and the endpoints of the interval, comparing these values to determine which is the largest and which is the smallest.