Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


f(t) = 12t − t³, −3 ≤ t < ∞

Verified step by step guidance

1. To find the local extrema of the function \( f(t) = 12t - t^3 \), we first need to find its critical points. This involves taking the derivative of the function and setting it equal to zero.
2. Calculate the derivative: \( f'(t) = \frac{d}{dt}(12t - t^3) = 12 - 3t^2 \).
3. Set the derivative equal to zero to find critical points: \( 12 - 3t^2 = 0 \). Solve for \( t \) to find the critical points.
4. Solve the equation \( 12 - 3t^2 = 0 \) to get \( t^2 = 4 \), which gives \( t = 2 \) and \( t = -2 \). These are the critical points within the domain \( -3 \leq t < \infty \).
5. Evaluate the function \( f(t) \) at the critical points and endpoints of the domain to determine the local extrema. Compare the values of \( f(t) \) at \( t = -3, -2, \) and \( 2 \) to identify the local maximum and minimum values.

Key Concepts

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are potential locations for local extrema, as they indicate where the function's slope changes direction. To find critical points, compute the derivative of the function and solve for values of the variable where the derivative equals zero or does not exist.

First Derivative Test

The First Derivative Test helps determine whether a critical point is a local maximum or minimum. By analyzing the sign of the derivative before and after the critical point, one can infer the behavior of the function. If the derivative changes from positive to negative, the point is a local maximum; if it changes from negative to positive, it is a local minimum.

Domain Considerations

Understanding the domain of a function is crucial when identifying extrema, as it defines the range of input values to consider. In this problem, the domain is given as −3 ≤ t < ∞, meaning the function is evaluated from t = -3 onwards. This affects where extrema can occur, especially at the boundary points, which must be checked separately.