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) = x¹ᐟ³(x + 8)

Verified step by step guidance

1. To determine where the function \( f(x) = x^{1/3}(x + 8) \) is increasing or decreasing, first find its derivative \( f'(x) \). Use the product rule: if \( u(x) = x^{1/3} \) and \( v(x) = x + 8 \), then \( f'(x) = u'(x)v(x) + u(x)v'(x) \).
2. Calculate \( u'(x) \) and \( v'(x) \). For \( u(x) = x^{1/3} \), use the power rule to find \( u'(x) = \frac{1}{3}x^{-2/3} \). For \( v(x) = x + 8 \), \( v'(x) = 1 \).
3. Substitute \( u'(x) \), \( v(x) \), \( u(x) \), and \( v'(x) \) into the product rule formula: \( f'(x) = \frac{1}{3}x^{-2/3}(x + 8) + x^{1/3}(1) \). Simplify this expression to find \( f'(x) \).
4. Find the critical points by setting \( f'(x) = 0 \) and solving for \( x \). These points, along with any points where \( f'(x) \) is undefined, will help determine intervals of increase and decrease.
5. Analyze the sign of \( f'(x) \) on the intervals determined by the critical points. If \( f'(x) > 0 \), the function is increasing on that interval; if \( f'(x) < 0 \), it is decreasing. Use this information to identify any local extrema by checking the sign changes of \( f'(x) \).

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 intervals of increase or decrease.

First Derivative Test

The First Derivative Test is used to determine whether a critical point is a local maximum, minimum, or neither. By analyzing the sign of the derivative before and after a critical point, one can ascertain the behavior of the function: if the derivative changes from positive to negative, the point is a local maximum; if it changes from negative to positive, it's a local minimum.

Interval Analysis

Interval analysis involves examining the sign of the derivative over different intervals to determine where the function is increasing or decreasing. By testing points within each interval, one can confirm the behavior of the function, ensuring a comprehensive understanding of its overall shape and identifying any local extrema.