Question

Identifying ExtremaIn 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

Accepted Answer

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 $$ f'(x) . $$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. 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 $$ f'(x) . If$$ $$ f'(x) \u003e 0 $$, the function is increasing; if $$ f'(x) \u003c 0 $$, the function is decreasing. 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. 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.

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 video answer for a similar problem:

Key Concepts

Derivative and Critical Points

Intervals of Increase and Decrease

Local Extrema