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

검증된 단계별 안내

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.

Find the first derivative \(f'(x)\) to determine where the function is increasing or decreasing. Use the power rule and chain rule: \(f'(x) = \frac{1}{3} \cdot 4(x+3)^3 = \frac{4}{3}(x+3)^3\).

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.

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).

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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