First Derivative Test


a. Locate the critical points of f.
b. Use the First Derivative Test to locate the local maximum and minimum values.
c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).


f(x) = x² + 3 on [-3,2]

Verified step by step guidance

1. To find the critical points of the function f(x) = x² + 3, first compute the derivative f'(x). The derivative of f(x) = x² + 3 is f'(x) = 2x.
2. Set the derivative equal to zero to find the critical points: 2x = 0. Solve for x to get x = 0. This is the critical point.
3. Use the First Derivative Test to determine if the critical point is a local maximum or minimum. Evaluate the sign of f'(x) around x = 0. For x < 0, f'(x) is negative, and for x > 0, f'(x) is positive, indicating a local minimum at x = 0.
4. To find the absolute maximum and minimum values on the interval [-3, 2], evaluate the function f(x) at the critical point and the endpoints of the interval. Calculate f(-3), f(0), and f(2).
5. Compare the values of f(-3), f(0), and f(2) to determine the absolute maximum and minimum values on the interval [-3, 2]. The smallest value is the absolute minimum, and the largest value is the absolute maximum.

Key Concepts

Critical Points

Critical points of a function occur where its derivative is either zero or undefined. These points are essential for identifying local extrema, as they represent potential locations where the function's behavior changes. To find critical points, one must first compute the derivative of the function and solve for values of x that satisfy the derivative equation.

First Derivative Test

The First Derivative Test is a method used to determine whether a critical point is a local maximum, local minimum, or neither. By analyzing the sign of the derivative before and after the critical point, one can conclude that if the derivative changes from positive to negative, a local maximum exists; if it changes from negative to positive, a local minimum exists.

Absolute Maximum and Minimum

The absolute maximum and minimum values of a function on a closed interval are the highest and lowest values the function attains within that interval, including at the endpoints. To find these values, one must evaluate the function at the critical points and the endpoints of the interval, comparing these values to determine which is the largest and which is the smallest.