Problem 33 - Calculus of a Single Variable

Question

Using the Second Derivative Test In Exercises \(31-42\) , find all relative extrema. Use the Second Derivative Test where applicable. $$ f(x)=x^{3}-3 x^{2}+3 $$

Step-by-Step Solution

Answer

The given function \(f(x) = x^3 - 3x^2 + 3\) has a local maximum at \(x = 0\) and a local minimum at \(x = 2\).

Step 1: Calculate the First Derivative

Start by calculating the first derivative of the given function. The derivative of \(x^3\) is \(3x^2\), the derivative of \(-3x^2\) is \(-6x\), and the derivative of 3 is 0. Thus, the first derivative \(f'(x)\) is:\[ f'(x) = 3x^2 - 6x \]

Step 2: Find Critical Points

Next, identify the critical points by setting the first derivative equal to zero and then solving for \(x\):\[ 3x^2 - 6x = 0 \]Factor out \(3x\), get:\[ 3x(x-2) = 0 \]Which yields \(x = 0\) and \(x = 2\) as critical points.

Step 3: Calculate the Second Derivative

The second derivative \(f''(x)\) is obtained by differentiating \(f'(x)\). The derivative of \(3x^2\) is \(6x\) and the derivative of \(-6x\) is \(-6\) hence:\[ f''(x) = 6x - 6\]

Step 4: Apply the Second Derivative Test

Evaluate the second derivative at the critical points. If \(f''(x)\) is positive at a critical point, then \(f(x)\) has a local minimum there. If \(f''(x)\) is negative, then \(f(x)\) has a local maximum there.For \(x = 0\):\[ f''(0) = 6*0 - 6 = -6 \]Because \(f''(0)\) is negative, \(f(x)\) has a local maximum at \(x = 0\).For \(x = 2\):\[f''(2) = 6*2 - 6 = 6 \]Because \(f''(2)\) is positive, \(f(x)\) has a local minimum at \(x = 2\).

Key Concepts

Critical Points

Critical points play a pivotal role in understanding the behavior of a function. These are specific points on the graph of a function where the slope is zero or the derivative does not exist. In simpler terms, if you imagine traveling along the curve, a critical point is where you'd either stop ascending or descending, or the path would be so steep that it's not clearly defined whether you're going up or down.

To find critical points, calculate the first derivative of the function and look for the values of the variable where this derivative equals zero or is undefined. These points are potential candidates for where the function may achieve its maximum or minimum values, or change its increasing-decreasing behavior.

First Derivative

The first derivative of a function can be thought of as the slope of the function at any given point. It tells you how fast the function is changing at that point. When the first derivative is positive, the function is increasing, and when it is negative, the function is decreasing.

By finding where the first derivative is zero, you can identify critical points which might indicate peaks, troughs, or plateaus in the function's graph. This is why computing the first derivative is an essential step in finding relative extrema of the function.

Second Derivative

While the first derivative provides information about the slope or rate of change, the second derivative sheds light on the curvature of the function. It tells us whether the function curves upwards or downwards at a particular point.

A positive second derivative indicates that the function is concave up and has a 'smiley face' shape, suggesting a local minimum. Conversely, a negative second derivative implies the function is concave down, shaped like a 'frowny face', and so might have a local maximum. The second derivative test, which involves evaluating the second derivative at critical points, is a reliable method to determine the nature of these points.

Relative Extrema

Relative extrema refer to the local maximums and minimums of a function - the highest or lowest points within a specific interval. These points are 'local' because they describe how the function behaves in the immediate vicinity of those points, rather than the overall highest or lowest points across its entire domain.

Tools like the first and second derivative tests help us identify these relative extrema by analyzing the slopes and concavities around the critical points. Knowledge of relative extrema is crucial since they often provide insight into the most important features of the function's graph, such as turning points and intervals of increase or decrease.

Local Maximum and Minimum

Local maximum and minimum values refer to the highest and lowest points in the vicinity of a given point on the graph. A local maximum is a point where the function changes from increasing to decreasing, meaning you’ve reached the top of the hill. A local minimum, on the other hand, is where the function transitions from decreasing to increasing – the bottom of a valley.

Through the derivative tests, we can determine not only where these points occur (critical points) but also classify them using the second derivative test. This test is especially useful because it provides a definitive answer about the concavity at the critical points, thus confirming whether they are indeed local maxima or minima.