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


h(t) = 2 + cos 2t on [0,π]

Verified step by step guidance

1. To determine the concavity of the function h(t) = 2 + cos(2t), we first need to find its second derivative. Start by finding the first derivative h'(t). The derivative of cos(2t) is -2sin(2t) using the chain rule.
2. Next, find the second derivative h''(t). Differentiate h'(t) = -2sin(2t) to get h''(t) = -4cos(2t) using the chain rule again.
3. To find intervals of concavity, set the second derivative equal to zero and solve for t: -4cos(2t) = 0. This simplifies to cos(2t) = 0. Solve for t in the interval [0, π].
4. The solutions to cos(2t) = 0 are t = π/4 and t = 3π/4 within the interval [0, π]. These are potential inflection points where the concavity might change.
5. Test the intervals (0, π/4), (π/4, 3π/4), and (3π/4, π) by choosing test points in each interval and evaluating the sign of h''(t). If h''(t) > 0, the function is concave up; if h''(t) < 0, the function is concave down. Identify the intervals of concavity and any inflection points based on these tests.

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 points of inflection.

Second Derivative Test

The second derivative test is a method used to determine the concavity of a function. By calculating the second derivative of a function, we can assess whether it is positive or negative over specific intervals. If the second derivative changes sign, it indicates a point of inflection, where the function changes from concave up to concave down or vice versa. This test is essential for identifying concavity and inflection points.

Inflection Points

Inflection points are points on a curve where the concavity changes. At these points, the second derivative of the function is either zero or undefined. Identifying inflection points is crucial for understanding the overall shape of the graph and how it behaves as it transitions between concave up and concave down. These points can provide valuable insights into the function's behavior and are often of interest in optimization problems.