Question

Explain how the Second Derivative Test is used.

Short answer

Answer: The Second Derivative Test is a method to determine the nature of critical points (local maxima, local minima, or saddle points) of a function. To apply the test: 1. Find the first derivative (f'(x)) and locate the critical points by setting f'(x) equal to 0 or finding points where f'(x) is undefined. 2. Find the second derivative (f''(x)). 3. Evaluate f''(x) for each critical point. - If f''(x) > 0, the critical point is a local minimum (concave up). - If f''(x) < 0, the critical point is a local maximum (concave down). - If f''(x) = 0 or undefined, the test is inconclusive, and another method like the First Derivative Test should be used to determine the critical point's nature.

Step-by-step solution

Understand Critical Points

A critical point of a function is any point on the graph of the function where the first derivative is either equal to zero or does not exist. These points are important because they can be potential local maximums, local minimums, or saddle points.

Find the Critical Points

To find the critical points of a given function, first, find the first derivative of the function. Then, set the first derivative equal to zero or find points where the derivative is undefined to find the critical points.

Understand the Second Derivative Test

The Second Derivative Test is a method to determine the nature of the critical points (local maxima, local minima, or saddle points). It involves evaluating the second derivative of the function at the critical points. The second derivative provides information about the concavity of the graph, which can help determine the type of critical point.

Apply the Second Derivative Test

Given a function f(x), apply the following steps of the Second Derivative Test: 1. Find the first derivative, f'(x), and locate the critical points by finding the values of x for which f'(x) = 0 or is undefined. 2. Find the second derivative, f''(x). 3. For each critical point, evaluate the second derivative, f''(x), at that point. The test outcomes can be interpreted as follows: - If f''(x) > 0 at a critical point, the function is concave up at that point, and the critical point is a local minimum. - If f''(x) < 0 at a critical point, the function is concave down at that point, and the critical point is a local maximum. - If f''(x) = 0 or is undefined at a critical point, the test is inconclusive, and one should resort to another method, such as the First Derivative Test, to determine the nature of the critical point.

Key concepts

Understanding Critical Points

Imagine driving along a hilly road; critical points are akin to the very top of a hill or the very bottom of a valley where your car would come to a temporary halt before ascending or descending. In mathematical terms, critical points occur where the slope of a function's graph—or its first derivative—is zero or undefined. These points are pivotal in analyzing functions because they potentially signify where the function reaches its highest or lowest value in a local area, known as local maxima and minima, or, less frequently, saddle points, which are neither.

Concavity and the Second Derivative

The concavity of a graph tells us the direction in which it curves. A function is concave up if it curves upwards, resembling a happy smile. Conversely, a function is concave down if it looks like a frown. The second derivative of a function, when evaluated at a particular point, informs us about the graph's concavity at that point. If the second derivative is positive, the graph is concave up. If negative, it's concave down. This attribute is crucial because it helps determine the nature of the critical points unearthed in the function's graph; whether they're positioned at the summit of a hill (local maxima) or in the depth of a valley (local minima).

Local Maxima and Minima

Local maxima and minima represent the peaks and troughs found in various segments of a function's graph. A local maximum is the highest point within a particular range, functioning as a ceiling that the function doesn't breach. Conversely, a local minimum is like the floorboard of a small region in the graph, being the lowest point the function reaches before it starts climbing up again. The second derivative test is especially handy when identifying if a critical point is a local maxima or minima. If the function curves up and forms a valley at the critical point, it's a local minima, but if it curves down and forms a hilltop, you've found a local maxima. These local extreme values are vital in various applications, such as optimization problems where one needs to find the best solution, be it a maximum profit or a minimum cost.