5.4 Using the First Derivative Test to Determine Relative (Local) Extrema

The First Derivative Test is a powerful tool in calculus that allows us to determine the relative (local) extrema of a function. A relative (local) extremum is a point on the graph of a function where the function reaches a maximum or a minimum value in a certain interval. To use the First Derivative Test, we first need to find the critical points of the function. A critical point is a point where the derivative of the function is equal to 0 or is not defined. Once we have identified the critical points, we need to find the sign of the derivative of the function on either side of the critical point. If the derivative is positive on one side of the critical point and negative on the other side, then the function is changing from increasing to decreasing at that point, indicating that the point is a relative maximum. If the derivative is negative on one side of the critical point and positive on the other side, then the function is changing from decreasing to increasing at that point, indicating that the point is a relative minimum. If the derivative is positive on both sides or negative on both sides, then the function is either always increasing or always decreasing at that point, and there is no relative extremum.

Example 1: Consider the function f(x) = x^3. The derivative of this function is f'(x) = 3x^2. The critical points are x = 0. To find the relative extrema, we need to find the sign of the derivative on either side of x = 0. Since 3x^2 is positive for all x, the function is always increasing at x = 0, so there is no relative extremum at this point. Example 2: Consider the function f(x) = x^4 - 2x^2. The derivative of this function is f'(x) = 4x^3 - 4x. The critical points are x = 0 and x = +/- sqrt(2)/2. To find the relative extrema, we need to find the sign of the derivative on either side of x = 0 and x = +/- sqrt(2)/2. Since 4x^3 - 4x is positive for x < -sqrt(2)/2 and x > sqrt(2)/2, and negative for -sqrt(2)/2 < x < sqrt(2)/2, the function is increasing on the interval (-infinity,-sqrt(2)/2) and (sqrt(2)/2,infinity) and decreasing on the interval (-sqrt(2)/2,sqrt(2)/2) The critical point x = 0 is a relative maximum and x = +/- sqrt(2)/2 is relative minimum. Example 3: Consider the function f(x) = 1/x. The derivative of this function is f'(x) = -1/x^2. The critical point is x = 0. To find the relative extrema, we need to find the sign of the derivative on either side of x = 0. Since -1/x^2 is negative for x < 0 and positive for x > 0, the function is decreasing on the interval (-infinity,0) and increasing on the interval (0,infinity). Therefore, the critical point x = 0 is a relative minimum. Example 4: Consider the function f(x) = x^3 + 3x^2 -3x. The derivative of this function is f'(x) = 3x^2 + 6x - 3. The critical points are x = -1 and x = -1/3. To find the relative extrema, we need to find the sign of the derivative on either side of x = -1 and x = -1/3. Since 3x^2 + 6x - 3 is negative for x < -1 and x > -1/3, and positive for -1 < x < -1/3, the function is decreasing on the interval (-infinity,-1) and ( -1/3,infinity) and increasing on the interval (-1,-1/3). The critical point x = -1 is a relative maximum and x = -1/3 is relative minimum.

It's important to note that the First Derivative Test only applies to relative extrema, and not global extrema. A global extremum is the highest or lowest value of a function over the entire domain of the function. To determine global extrema, additional techniques such as the Second Derivative Test or the Extreme Value Theorem may be needed.

Additionally, it's important to graph the function and visually inspect the critical points to confirm the results obtained from the First Derivative Test, as the test only gives information about the relative extrema in a small interval around the critical point.

In conclusion, the First Derivative Test is a powerful tool in calculus that allows us to determine relative extrema of a function by identifying critical points and analyzing the sign of the derivative on either side of those points. It's important to keep in mind the limitations of the test and use additional techniques to determine global extrema.