Identifying Extrema

In Exercises 19–40:

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

f(r) = 3r³ + 16r

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First, find the derivative of the function f(r) = 3r³ + 16r. The derivative, f'(r), will help us determine where the function is increasing or decreasing.

Calculate the derivative: f'(r) = d/dr (3r³ + 16r). Use the power rule for differentiation, which states that d/dr (r^n) = n*r^(n-1).

Apply the power rule: f'(r) = 3 * 3r² + 16 * 1 = 9r² + 16.

Set the derivative f'(r) = 9r² + 16 equal to zero to find critical points, which are potential points where the function changes from increasing to decreasing or vice versa.

Solve the equation 9r² + 16 = 0 for r. These critical points will help determine the intervals of increase and decrease. Analyze the sign of f'(r) around these points to identify the intervals where the function is increasing or decreasing.

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주요 개념

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Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are essential for determining where a function changes from increasing to decreasing or vice versa. For the function f(r) = 3r³ + 16r, finding the derivative and setting it to zero will help identify these critical points.

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 each critical point, we can ascertain the intervals where the function is increasing or decreasing. This test is crucial for understanding the behavior of f(r) = 3r³ + 16r around its critical points.

Increasing and Decreasing Intervals

A function is increasing on an interval if its derivative is positive throughout that interval, and decreasing if its derivative is negative. By applying the First Derivative Test to f(r) = 3r³ + 16r, we can identify these intervals, which are key to understanding the overall shape and behavior of the function.

Identifying ExtremaIn Exercises 19–40:a. Find the open intervals on which the function is increasing and those on which it is decreasing.f(r) = 3r³ + 16r