Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. See Example 1. ƒ(x)=1/3(x+3)^4-3

Verified step by step guidance

1. Identify the function: \( f(x) = \frac{1}{3}(x+3)^4 - 3 \). This is a transformed quartic function.
2. Find the derivative \( f'(x) \) to determine where the function is increasing or decreasing. Use the power rule: \( f'(x) = \frac{d}{dx}\left(\frac{1}{3}(x+3)^4 - 3\right) \).
3. Calculate the derivative: \( f'(x) = \frac{4}{3}(x+3)^3 \).
4. Set \( f'(x) = 0 \) to find critical points: \( \frac{4}{3}(x+3)^3 = 0 \). Solve for \( x \).
5. Analyze the sign of \( f'(x) \) around the critical points to determine intervals of increase and decrease. If \( f'(x) > 0 \), the function is increasing; if \( f'(x) < 0 \), it is decreasing.

Key Concepts

Function Graphing

Graphing a function involves plotting its points on a coordinate plane to visualize its behavior. This includes identifying key features such as intercepts, turning points, and asymptotes. Understanding how to graph polynomial functions, like the given function, is essential for analyzing their increasing and decreasing intervals.

Increasing and Decreasing Intervals

An increasing interval of a function is where the function's output values rise as the input values increase, while a decreasing interval is where the output values fall. To determine these intervals, one must analyze the first derivative of the function, which indicates the slope. Positive values of the derivative suggest increasing behavior, while negative values indicate decreasing behavior.

Critical Points

Critical points are values of the independent variable where the derivative of the function is zero or undefined. These points are crucial for determining where a function changes from increasing to decreasing or vice versa. By evaluating the function at these points, one can identify the intervals of increase and decrease effectively.