Section 4.1: Graphing and Optimization

First Derivative Test: Increasing and Decreasing Functions

The first derivative test is a fundamental tool in calculus for determining where a function is increasing or decreasing, and for identifying local minimum and maximum points. This test uses the sign of the derivative, f'(x), to analyze the behavior of the function f(x) on an interval.

• Increasing Function: A function f(x) is increasing on an interval (a, b) if for any x_1 < x_2 in (a, b), f(x_1) < f(x_2). If f'(x) > 0 for all x in (a, b), then f(x) is increasing on that interval.

• Decreasing Function: A function f(x) is decreasing on an interval (a, b) if for any x_1 < x_2 in (a, b), f(x_1) > f(x_2). If f'(x) < 0 for all x in (a, b), then f(x) is decreasing on that interval.

Example: Consider the graph below. The intervals where the function is increasing or decreasing can be identified by observing where the slope is positive (increasing) or negative (decreasing).

Local Minimums and Maximums (Relative Extrema)

A local maximum of f(x) at x = c occurs if there is an interval (a, b) containing c such that f(x) \leq f(c) for all x in (a, b). A local minimum occurs if f(x) \geq f(c) for all x in (a, b). These points are also called local extrema or turning points.

• Local Maximum: Occurs where the function changes from increasing to decreasing (f'(x) changes from positive to negative).

• Local Minimum: Occurs where the function changes from decreasing to increasing (f'(x) changes from negative to positive).

Example: On the graph below, local maxima and minima are found at the peaks and valleys, respectively.

Applying the First Derivative Test to a Function

To find intervals of increase/decrease and local extrema for a given function, follow these steps:

1. Find the first derivative f'(x).

2. Solve f'(x) = 0 to find critical points.

3. Test intervals between critical points to determine the sign of f'(x).

4. Classify each critical point as a local minimum, maximum, or neither using the sign changes of f'(x).

Example: For the function f(x) = 2x^3 + 3x^2 - 120x:

• Compute the derivative:

• Set f'(x) = 0 and solve for x to find critical points.

• Test intervals around these points to determine where f(x) is increasing or decreasing.

Practice Problems

Apply the first derivative test to the following functions to find intervals of increase/decrease and local extrema:

Additional info: For each quadratic function, the vertex represents the local extremum. For ax^2 + bx + c, the vertex is at .