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

Verified step by step guidance

1. Identify the function given: \(f(x) = \frac{1}{3}(x+3)^4 - 3\). This is a polynomial function involving a quartic term shifted horizontally and vertically.
2. Find the first derivative \(f'(x)\) to determine where the function is increasing or decreasing. Use the power rule and chain rule: \(f'(x) = \frac{1}{3} \cdot 4(x+3)^3 = \frac{4}{3}(x+3)^3\).
3. Set the derivative equal to zero to find critical points: \(\frac{4}{3}(x+3)^3 = 0\). Solve for \(x\) to find critical values where the function could change from increasing to decreasing or vice versa.
4. Analyze the sign of \(f'(x)\) on intervals determined by the critical points. Since the derivative is a cubic expression, test values less than and greater than the critical point \(x = -3\) to see if \(f'(x)\) is positive (increasing) or negative (decreasing).
5. Based on the sign analysis, conclude the largest open intervals where \(f(x)\) is increasing or decreasing. Remember, if \(f'(x) > 0\) on an interval, \(f\) is increasing there; if \(f'(x) < 0\), \(f\) is decreasing.

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For polynomial and polynomial-like functions such as ƒ(x) = (1/3)(x+3)^4 - 3, the domain is all real numbers since there are no restrictions like division by zero or square roots of negative numbers.

Increasing and Decreasing Intervals

A function is increasing on an interval if its output values rise as x increases, and decreasing if its output values fall as x increases. Identifying these intervals involves analyzing the function's behavior, often using the first derivative to determine where the slope is positive (increasing) or negative (decreasing).

Using the First Derivative to Analyze Function Behavior

The first derivative of a function gives the rate of change or slope at any point. By finding where the derivative is zero or undefined, we locate critical points that help divide the domain into intervals. Testing the sign of the derivative on these intervals reveals where the function is increasing or decreasing.