Identifying Extrema


In Exercises 19–40:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local extreme values, if any, saying where they occur.


f(x) = (x² − 3) / (x − 2), x ≠ 2

Verified step by step guidance

1. To determine where the function \( f(x) = \frac{x^2 - 3}{x - 2} \) is increasing or decreasing, first find its derivative \( f'(x) \). Use the quotient rule: \( f'(x) = \frac{(x - 2)(2x) - (x^2 - 3)(1)}{(x - 2)^2} \). Simplify the expression to find \( f'(x) \).
2. Set \( f'(x) = 0 \) to find critical points. Solve the equation \( (x - 2)(2x) - (x^2 - 3) = 0 \) to find the values of \( x \) where the derivative is zero. These are potential points where the function changes from increasing to decreasing or vice versa.
3. Determine the sign of \( f'(x) \) on the intervals defined by the critical points and the point where the function is undefined (\( x = 2 \)). Choose test points in each interval to evaluate the sign of \( f'(x) \). If \( f'(x) > 0 \), the function is increasing; if \( f'(x) < 0 \), the function is decreasing.
4. Identify the local extrema by evaluating \( f(x) \) at the critical points found in step 2. A change from positive to negative in \( f'(x) \) indicates a local maximum, while a change from negative to positive indicates a local minimum.
5. Summarize the intervals of increase and decrease, and list any local extrema along with their locations. Remember to exclude \( x = 2 \) from the domain of the function, as it is undefined there.

Key Concepts

Derivative and Critical Points

The derivative of a function provides information about its rate of change. To find where a function is increasing or decreasing, calculate its derivative and identify critical points where the derivative is zero or undefined. These points help determine intervals of increase or decrease by testing the sign of the derivative in each interval.

Intervals of Increase and Decrease

Once critical points are identified, determine the intervals on which the function is increasing or decreasing. This involves testing the sign of the derivative in each interval between critical points. A positive derivative indicates the function is increasing, while a negative derivative indicates it is decreasing.

Local Extrema

Local extrema refer to the local maximum or minimum values of a function. These occur at critical points where the derivative changes sign. A change from positive to negative indicates a local maximum, while a change from negative to positive indicates a local minimum. Evaluating the function at these points confirms the extrema.