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) = x − 6√(x − 1)

Verified step by step guidance

1. To determine where the function \( f(x) = x - 6\sqrt{x - 1} \) is increasing or decreasing, first find its derivative \( f'(x) \). Use the power rule and chain rule to differentiate: \( f'(x) = 1 - \frac{6}{2\sqrt{x - 1}} \). Simplify this to \( f'(x) = 1 - \frac{3}{\sqrt{x - 1}} \).
2. Identify the domain of \( f(x) \). Since \( \sqrt{x - 1} \) is defined for \( x \geq 1 \), the domain of \( f(x) \) is \( x > 1 \).
3. Find the critical points by setting \( f'(x) = 0 \). Solve \( 1 - \frac{3}{\sqrt{x - 1}} = 0 \) to find the critical points. This simplifies to \( \sqrt{x - 1} = 3 \), leading to \( x - 1 = 9 \), so \( x = 10 \).
4. Determine the intervals of increase and decrease by testing points in the intervals \( (1, 10) \) and \( (10, \infty) \). Choose a test point in each interval, such as \( x = 2 \) and \( x = 11 \), and evaluate \( f'(x) \) at these points to determine the sign of the derivative.
5. Identify the local extrema by analyzing the sign changes of \( f'(x) \). If \( f'(x) \) changes from positive to negative at \( x = 10 \), then \( f(x) \) has a local maximum at \( x = 10 \). If it changes from negative to positive, it would be a local minimum. Evaluate \( f(x) \) at \( x = 10 \) to find the local extreme value.

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 are potential locations for local extrema and help determine the behavior of the function on different intervals.

Increasing and Decreasing Intervals

A function is increasing on an interval if its derivative is positive throughout that interval, and decreasing if the derivative is negative. By analyzing the sign of the derivative on intervals between critical points, you can determine where the function is rising or falling, which is essential for identifying local extrema.

Local Extrema

Local extrema are points where a function reaches a local maximum or minimum value. These occur at critical points where the derivative changes sign. To confirm whether a critical point is a local maximum or minimum, use the first or second derivative test, which examines the behavior of the derivative around these points.