Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.


f(x) = ³√(x - 4)

Verified step by step guidance

1. First, find the first derivative of the function f(x) = ³√(x - 4). This involves using the chain rule. The derivative of ³√u with respect to u is (1/3)u^(-2/3). Therefore, f'(x) = (1/3)(x - 4)^(-2/3).
2. Next, find the second derivative, f''(x), to determine concavity. Differentiate f'(x) = (1/3)(x - 4)^(-2/3) again using the chain rule. The derivative of u^(-2/3) is (-2/3)u^(-5/3), so f''(x) = (-2/9)(x - 4)^(-5/3).
3. Determine the intervals of concavity by analyzing the sign of f''(x). The function is concave up where f''(x) > 0 and concave down where f''(x) < 0. Since f''(x) = (-2/9)(x - 4)^(-5/3), it is important to note that the sign of f''(x) depends on the sign of (x - 4).
4. Identify any inflection points by finding where f''(x) changes sign. This occurs when x = 4, as the expression (x - 4) changes from negative to positive or vice versa.
5. Summarize the findings: The function is concave down for x < 4 and concave up for x > 4. The point x = 4 is an inflection point where the concavity changes.

Key Concepts

Concavity

Concavity refers to the direction in which a function curves. A function is concave up on an interval if its second derivative is positive, indicating that the slope of the tangent line is increasing. Conversely, it is concave down if the second derivative is negative, meaning the slope is decreasing. Understanding concavity helps in analyzing the behavior of functions and identifying intervals of increase or decrease.

Second Derivative Test

The second derivative test is a method used to determine the concavity of a function and locate inflection points. By calculating the second derivative of a function, we can assess where it changes sign, indicating a transition between concave up and concave down. Inflection points occur where the second derivative is zero or undefined, and the concavity changes, providing critical insights into the function's graph.

Inflection Points

Inflection points are specific points on a curve where the concavity changes. At these points, the function transitions from being concave up to concave down or vice versa. Identifying inflection points is essential for sketching the graph of a function accurately, as they indicate where the curvature of the graph alters, affecting the overall shape and behavior of the function.